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Second-order Processes

2011-08-15T22:28:54Z

Chrisb: /* Kerr and Pockels Effects */

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[http://depts.washington.edu/cmditr/media/NLO_materials.html Concept Map for Second Order Non linear Optics]

Second order non linear optics involve the search for materials whose optical properties can be controlled with an applied electrical or optical field. The are second order because the effect is quadratic with respect to field strength. These extremely fast processes can be used for optical switching in telecommunication and the frequency effects can be used for specialized spectroscopy, imaging and scanning.

== Electro optical materials ==

=== EO Materials have a voltage-controlled index of refraction. ===
Light has a known speed in a vacuum. But when enters a material it slows down. Light has a electrical and magnetic component. The electrical component will interact with the charge distribution of the atom in the material is passed through. The interaction will slow the light down.

The index of refraction = speed of light in vacuum / speed of light in material.

An electro-optic material (in a device) permits electrical and optical signals to “talk” to each other through an “easily perturbed” electron distribution in the material. A low frequency (DC to 200 GHz) electric field (e.g., a television [analog] or computer [digital] signal) is used to perturb the electron distribution (e.g., p-electrons of an organic chromophore) and that perturbation alters the speed of light passing through the material as the electric field component of light (photons) interacts with the perturbed charge distribution.

Because the speed of light is altered by the application of a control voltage, electro-optic materials can be described as materials with a voltage-controlled index of refraction.

For example, you apply and electric field that alters the charge distribution of the material, which in turn influences the propagation of light through the material. (Pockels effect). The reverse process is called optical rectification. When there are two fields involved this is called a second-order nonlinear optical effect.

<div id="Flash">Electro Optic Effect Animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/eo_lightspeed.swf</swf>

In this Flash animation a light source emits photons which travel through the material at the speed of light. When there is no field the electro-optic material has no induced electron asymmetry. Click the battery to add an electric field. The EO materials change their electron distribution which changes their index of refraction so as to slow down light moving through the eo polymer. If these two light beams recombined their wave behavior might interfere. It is this property that can be used to modulate light.

=== Types of EO materials. ===

The response speed of EO materials relates to the mass of the entity that is moved.

'''Liquid Crystals''' -In liquid crystalline materials there is a change in molecular orientation, which changes the dipole moment and charge distribution of the material, which is turn changes the velocity of light moving through the material. This can be measured by the retardation of the speed of light measure in picometers per volt applied. This is a large effect (>10,000 picometers (pm)/V) but rather slow (103 -106 sec) because we are moving a lot of mass. This not so useful for high speed communication.

'''Inorganic crystals''' the electric field causes ion diplacement. This is a small effect (30pm/V) but faster (10-10 sec) because a smaller ion with less mass is moving.

'''electron chromophore polymer'''- A third technique uses π electron chromophore containing polymers and dendrimers. Electric field can change their π electron distribution. This has a large EO activity (>500 pm/v) and very fast into the terahertz (thz) region (10-14 sec).
 
[[Image:organic_modulation_speed.png|thumb|400px|The advantage of organic molecules is high frequency modulation.]]

Organic EO materials have the potential for faster response, lower drive voltage, larger bandwidth, lighter weight and lower cost. They can also be tailored to specific applications and integrated at the chip scale level.
 

== Polarization Effects ==
=== NLO Chromophore ===

[[Image:PASchromophore.JPG|thumb|300px|]]
The basic unit of organic electro-optics is the EO-active material, or chromophore.

This chromophore can be thought of as a molecular oscillator interacting with EM radiation.

Electron donor and acceptor moeties are connected by a π -conjugated bridge that serves as a conduit for electron density.

 
=== Asymmetric Polarization ===
[[Image:4-nitroaniline.png|thumb|300px|4-nitroaniline]]

In second order non linear optics we are concerned with asymmetric polarization of light absorbing molecules in a material.

[[Image:Nlo_effect.png|thumb|500px|Linear and nonlinear polarization response to electric field]]

The diagram is a representation of what happens to a molecule that is asymmetric when an electric field is applied. A molecule with a dipole such as 4-nitroaniline has a charge distribution that leads to a dipole. One side is a donor (d) and an acceptor (a) with a π conjugated system. The magnitude of the induced dipole will be greatest when the electric field is aligned so as to move the electron density towards the electron donor end of the molecule. In a symmetric molecule is there a linear polarizability shown as the straight line. The greater the charge, the greater the induced dipole. In an asymmetric material there a nonlinear effect which makes it easier to polarize in one direction than the other, and increasing electric field has an exponentially increasing effect.

In the presence of an oscillating electric field a linear material will have an induced dipole that is in phase and has the same frequency as the applied field.
 
[[Image:Polarizationwave.png|thumb|300px|An asymmetric polarization response to a symmetric oscillating field]]

The application of a symmetric field (i.e. the electric field associated with the light wave) to the electrons in an anharmonic potential leads to an asymmetric polarization response. This polarization wave has flatted troughs (diminished maxima) in one direction and sharper and higher peaks (accentuated maxima) in the opposite direction, with respect to a normal sine wave.

It is possible to find the sum of waves that would result in such a wave using techniques such as fourier transform. In the case of a symmetric polarization it is simply the sine wave of the applied field.

This asymmetric polarization can be Fourier decomposed (deconvolved) into a static DC polarization component with components at the fundamental frequency superimposed with a second harmonic frequency (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluorescence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/07 Assymetric Polarization.swf</swf>

=== Fourier Analysis of Asymmetric Polarization Wave ===
[[Image:Fourier_harmonics.png|thumb|300px|Combining a fundamental wave and a second harmonic to get a complex polarization wave]]
This asymmetric polarization can be Fourier decomposed (deconvoluted) into a static DC polarization component and components at the fundamental frequency superimposed with a second harmonic frequencies (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluoresence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

=== Expression for Microscopic Nonlinear Polarizabilities ===

The linear polarizability is the first derivative of the dipole moment with respect to electric field.
In non-linear optical effects the plot of induced polarization vs applied field can be corrected using higher corrections with a Taylor series expansion, including the second derivative of the dipole moment with respect to electric field times the field squared with a single electric field, or higher order terms using the third derivative of dipole moment vs field the field cubed. Μ is the total dipole moment in the molecule which is a sum of the static dipole plus several field dependent term.

:<math>\mu = \mu_0 + (\partial \mu_i / \partial E_j)_{E_0}E_j \quad + \quad 1/2 (\partial^2 \mu_i / \partial E_jE_k)_{E_0} E_jE_k \quad+ \quad 1/6(\partial^3\mu_i / \partial E_jE_kE_j)_{E_0} E_jE_kE_j\,\!</math>

Where
:<math>\mu\,\!</math> is the microscopic nonlinear polarizability
:<math>E_i E_k E_j\,\!</math> are the electric field (vectors)
:<math>\mu = \mu_0 \quad+\quad \alpha_{ij}E_j \quad+\quad \beta _{ijk}/ 2 E E \quad+\quad \gamma_{ijkl} / 6 E E E + ...\,\!</math>

Where:

:<math>\alpha\,\!</math> is linear polarizability
:<math>\beta\,\!</math> is the [[first hyperpolarizability]] ( a third rank tensor with 27 permutations although some are degenerate)
:<math>\gamma\,\!</math> is the [[second hyperpolarizability]], responsible for third order non linear optics.

The terms beyond αE are not linear (they have exponential terms) in E and are therefore referred to as the nonlinear polarization and give rise to nonlinear optical effects. Note that Ej, Ek, and Ej are vectors representing the direction of the polarization of the applied field with respect to the molecular coordinate frame. Molecules are asymmetric have different polarizabilities depending the direction of the applied electric field.

Alpha is the second derivative of the dipole moment with respect to field, and is also the first derivative of the polarizability with respect to field. Beta is the first derivative of polarizability with respect to field, and gamma is the first derivative of the first hyperpolarizability with respect to field.

Nonlinear polarization becomes more important with increasing field strength, since it scales with higher powers of the field (quadratic or cubic relationships). Second harmonic generation was not observed until 1961 after the advent of the laser. Under normal conditions,
:<math>\alpha_{ij}E \quad > \quad \beta_{ijk}/2 E·E \quad > \quad \gamma_{ijkl} /6 E·E·E.\,\!</math>

Thus, there were few observations of NLO effects with normal light before the invention of the laser with its associated large electric fields.

With very large electric fields there can be dielectric breakdown of the material.

The observed bulk polarization density is given by an
expression analogous to (7):

:<math>P = P_o + \chi^{(1)} ·E + \chi^{(2)}·· EE + \chi^{(3)}···EEE+ ...\,\!</math> (8)

where the :<math>\chi^{(i)}\,\!</math> susceptibility coefficients are tensors of order i+1 (e.g., :<math>\chi^{(2)}_{ijk}\,\!</math>).
Po is the intrinsic static dipole moment density of the sample.

The linear polarizability is the ability to polarize a molecule, the linear susceptibility is bulk polarization density in a materials which has to do with the polarizability of the molecules and the density of those molecules in the material. More molecules means a higher susceptibility.

=== Taylor Expansion for Bulk Polarization ===
Consider a simple molecule with all the fields being identical.

:<math>P = P_o + \chi^{(1)} ·E + 1/2\chi^{(2)}·· E^2 + 1/6\chi^{(3)}···E^3+ ...\,\!</math>

In a Taylor series expansion the dots refer the fact that these are tensor products. Just as a molecule can only have a non-zero beta if it is noncentrosymmetric, a material can only have a :<math>\chi^{(2)}\,\!</math> if the material is noncentrosymmetric (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules lead to zero :<math>\chi^{(2)}\,\!</math>) .

In a centrosymmetric material a perturbation by an electric field (E) leads to a polarization P. Therefore, application of an electric field (–E) must lead to a polarization –P.

Now consider the second order polarization in a centrosymmetric material.

:<math>P = \chi^{(2)}·· E^2,\,\!</math> (10)

:<math> –P = \chi^{(2)}·· (–E)^2 = \chi^{(2)}·· E^2\,\!</math> (11)

This only occurs when P = 0, therefore :<math>\chi^{(2)}\,\!</math> must be 0.

This means that if we use quantum mechanics to design molecules that will have large hyperpolarizabilities the effort will be wasted if the molecules arrange themselves in a centrosymmetric manner resulting in bulk susceptibility of zero. The design therefore must include both arranging for the desired electronic properties, but also configuring the molecule so that those molecules will not line up in a centrosymmetric manner in the material. A solution of molecules can also exhibit some centrosymmetry.

== Frequency Effects ==

=== Frequency Doubling and Sum-Frequency Generation ===

One nonlinear optical phenomena is that when you shine light at one frequency on a material you get out light with twice the frequency. This process is known as second harmonic generation (SHG). SHG is a special type of sum frequency generation (SFG). SFG occurs when ω3 = ω1 + ω2. If ω3 = ω1 – ω2, this results in difference frequency generation (DFG). These processes are described in more detail below.

The electronic charge displacement (polarization) induced by an oscillating electric field (e.g., light) can be viewed as a classical oscillating dipole that itself emits radiation at the oscillation frequency.

For linear first-order polarization, the radiation has the same frequency as the incident light.

=== Taylor Expansion with Oscillating Electric Fields-SHG ===

The electric field of a plane light wave can be expressed as

:<math>E = E_0 cos(\omega t)\,\!</math>

Using a power series expansion :<math>Ecos^2(\omega t) E\,\!</math> can be substituted for E

:<math>P = P_0 + \chi^{(1)}E_0 cos(\omega t) + \chi^{(2)} E_0^2cos^2(\omega t) + \chi^{(3)} E_0^3 cos^3(\omega t) + ...\,\!</math>

Where
:<math>P_0\,\!</math> is the static polarizablity

Since
:<math>cos^2(\omega t)\,\!</math> equals :<math>1/2 + 1/2 cos(2 \omega t)\,\!</math>,

the first three terms of equation (13) become:

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (14)

This is the origin of the process of optical rectification and second harmonic generation.

=== Second Harmonic Generation (SHG) ===

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (16)

Physically, equation (16) states that the polarization consists of a:

*Second-order DC field contribution to the static polarization (first term),

*Frequency component ω corresponding to the light at the incident frequency (second term) and

*A new frequency doubled component, :<math>2\omega\,\!</math> (third term)-- recall the asymmetric polarization wave and its Fourier analysis.

=== Sum and Difference Frequency Generation ===

In the more general case (in which the two fields are not constrained to be equal), NLO effects involves the interaction of NLO material with two distinct waves with electric fields E1 with the electrons of the NLO material.

Consider two laser beams E1 and E2, the second-order term of equation (4) becomes:
:<math>\chi^{(2)}·E_1cos(\omega_1t)E_2cos(\omega_2t)\,\!</math> (15)

From trigonometry we know that equation (15) is equivalent to:
:<math>1/2\chi^{(2)}·E_1E_2cos [(\omega_1 + \omega_2)t] +1/2\chi^{(2)}·E_1E_2cos [(\omega_1 - \omega_2)t]\,\!</math> (16)

Thus when two light beams of frequencies ω1 and ω2 interact in an NLO material, polarization occurs at sum :<math>(\omega_1 + \omega_2)\,\!</math> and difference :<math>(\omega_1 - \omega_2)\,\!</math> frequencies.

This electronic polarization will therefore, re-emit radiation at these frequencies.

The combination of frequencies is called sum (or difference) frequency generation (SFG) of which SHG is a special case. This is how a tunable laser works.

Note that a very short laser pulse will result in a band or distribution of frequencies due to the Heisenberg Uncertainty Principle. Those bands will add and subtract resulting in some light which is twice the frequency if they added, and some light that is very low frequency (0+ or – the difference), resulting from the difference between the frequencies. This is the process enabling Terahertz spectroscopy. Terahertz is very low frequency light.

Low frequency light is scattered less than high frequency light. For example if you look through a glass of milk there is “index inhomogeneity” in the milk due the presence of protein and fat. Terahertz radiation can be used for surveillance. A terahertz detector scanner will reveal materials that have different index of refraction.

== Electro-optic effects ==

=== Kerr and Pockels Effects ===

John Kerr and Friedrich Pockels discovered in 1875 and 1893, respectively, that the refractive index of a material could be changed by applying a DC or low frequency electric field. This are in fact non-linear optical effects but they often not thought of as such because they don’t require a laser.

Electric impermeability of a material can be expressed as:

:<math>n \equiv \frac {\epsilon_0}{\epsilon} = \frac{1}{n^2}\,\!</math>

:<math>\eta (E) = \eta + rE +SE^2\,\!</math>

Where
:<math>\epsilon_0\,\!</math> is the dielectric constant of free space
:<math>\epsilon\,\!</math> is the dielectric constant

'''Pockels effect'''
[[Image:Pockels_graph.png|thumb|200px|The Pockels effect has a linear relation to applied field]]
In the Pockels effect an applied electric field changes the refractive index of certain materials.

:<math>\eta(E) = \eta - \frac {1} {2}rn^3 E\,\!</math>

Where:

'''r''' is Pockels coefficient or Linear Electro-optic Coefficient, r~ 10-12 – 1—10 m/V, typically.

This is a linear function with respect to the electric field, the higher the r the greater the change. It is cubic with respect to the refractive index so materials with high intrinsic refractive indexes will change more. Some examples include NH4H2PO4(ADP), KH2PO4(KDP), LiNbO3, LiTaO3, CdTe
 
'''The Kerr effect'''
[[Image:Kerr_graph.png|thumb|200px|The Kerr effect has a parabolic relationship to applied field]]
:<math>\eta(E) = \eta – ½ Sn^3E^2\,\!</math>

Where

'''S''' is the Kerr coefficient
*S~ 10-18 – 10-14 mV in crystals
*S~ 10-22 – 10-19 mV in liquids

This is similar to the Pockels effect except that the refractive index varies parabolically or quadratically with the electric field.

This a process that occurs in second order nonlinear optical materials. It is a third order nonlinear optical process. Not all materials are second order nonlinear optical materials, only those that are centrosymmetric. However all materials have a χ(3) even if they are centrosymmetric.

It is possible to change the amplitude, phase or path of light at a given frequency by using a static DC electric field to polarize the material and modify the refractive indices. When light enters a material with a higher refractive index it is phase shifted and the waves become compressed. The direction is also changed. So by changing the refractive index it is possible to change the path of the light.

Consider the special case :<math>\omega_2 = 0\,\!</math> [equation (15)] in which a DC electric field is applied to the material.

The optical frequency polarization (Popt) arising from the second-order susceptibility is given by:

:<math>P_{opt} =\chi^{(2)}·E_1E_2(cos \omega_1 t)\,\!</math> (17)

where:
E1
E2 is the magnitude of the electric field caused by voltage applied to the nonlinear material (a voltage not optical frequency).

Recall that the refractive index is related to the linear susceptibility that is given by the second term of Equation (14):

:<math>\chi^{(1)}·E_1(cos \omega_1 t)\,\!</math> (18)

The total optical frequency polarization (Popt) is the χ(1) term plus the χ(2) term:

:<math>P_{opt} = \chi^{(1)}·E_1(cos_1t) +\chi^{(2)}·E_1E_2(cos \omega_1t)\,\!</math> (19)

Then factor out <math>E_1(cos \omega_1t)\,\!</math> :

:<math> P_{opt} = [\chi^{(1)} + \chi^{(2)}·E_2] E_1(cos \omega_1t)\,\!</math> (20)

χ(1) is linear susceptability which relates to the dielectric constant, which in turn relates to the square of the refractive index. A change in the linear susceptablity changes the index of refraction. The second term: χ(2) times the magnitude of the voltage (E2) means that the susceptability of the material, the dielectric constant of the material, and the refractive index of the material can be altered by changing the applied voltage.

You can shine light on second order nonlinear optical materials and get out different frequencies, or shine one laser beam, apply an electric field and then modulate the refractive index. For example, light can travel freely between two fibers that are very close to each other with the same refractive index. But if the fibers have a different refractive index light will stay in one fiber or the other.

By changing the refractive index you can move light from one fiber to another; it provides a means of switching light in waveguides.

'''Summary'''

*The applied field in effect changes the linear susceptibility and thus the refractive index of the material.

*This is, known as the linear electro-optic (LEO) or Pockels effect, and is used to modulate light by changing the applied voltage.

*At the atomic level, the applied voltage is anisotropically distorting the electron density within the material. Thus, application of a voltage to the material causes the optical beam to "see" a different material with a different polarizability and a different anisotropy of the polarizability than in the absence of the voltage.

*Since the anisotropy is changed upon application of an electric field, a beam of light can have its polarization state (i.e., ellipticity) changed by an amount related to the strength and orientation of the applied voltage, and travel at a different speed and possibly in a different direction.

=== Index modulation ===

Quantitatively, the change in the refractive index as a function of the applied electric field is approximated by
the general expression:

:<math>1/\underline{n}_{ij}2 = 1/n_{ij}2 + r_{ijk}E_k + s_{ijkl}E_kE_l + ... \,\!</math> (21)

where
:<math>\underline{n}_{ij}\,\!</math> are the induced refractive indices,
:<math>n_{ij}\,\!</math> is the refractive index in the absence of the electric field,

:<math>r_{ijk}\,\!</math> is the linear or Pockels coefficients, Δn for E = 106 V/m is 10-6 to 10-4 (crystals) and;

:<math>s_{ijkl}\,\!</math> are the quadratic or Kerr coefficients.

=== r coefficients ===

The optical indicatrix (that characterizes the anisotropy of the refractive index) therefore changes as the electric field within the sample changes. The of map the index of refraction with respect to each polarization of light produces a surface that looks something like a football. The electric field allows you to change the shape of the football.

Electro-optic coefficients are frequently defined in terms of rijk.
The "r" coefficients form a tensor (just as do the coefficient of α).

The subscripts ijk are the same as those used with β. The first subscript (i) refers to the resultant polarization of the material along a defined axis and the following subscripts j and k refer to the orientations of the applied fields, one is the optical frequency field and k is the voltage.

=== Applications of Electro-optic Devices ===
[[Image:Network.png|thumb|400px|EO materials can be used at many locations in a network]]
A network has a variety of devices that provide input from to a transmitter, connected by a electro-optic modulator (EOM) through a switching network, to a receiver with a photodetector, and then are connected to display devices. Nonlinear optical materials can be used for any of these applications. They can used to create terahertz radiation and to create specific wavelengths of light for spectroscopy.
 

[[category:non linear optics]]
[[category:second order NLO]]
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Second-order Processes

2011-08-12T23:34:13Z

Chrisb: /* Frequency Doubling and Sum-Frequency Generation */

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[http://depts.washington.edu/cmditr/media/NLO_materials.html Concept Map for Second Order Non linear Optics]

Second order non linear optics involve the search for materials whose optical properties can be controlled with an applied electrical or optical field. The are second order because the effect is quadratic with respect to field strength. These extremely fast processes can be used for optical switching in telecommunication and the frequency effects can be used for specialized spectroscopy, imaging and scanning.

== Electro optical materials ==

=== EO Materials have a voltage-controlled index of refraction. ===
Light has a known speed in a vacuum. But when enters a material it slows down. Light has a electrical and magnetic component. The electrical component will interact with the charge distribution of the atom in the material is passed through. The interaction will slow the light down.

The index of refraction = speed of light in vacuum / speed of light in material.

An electro-optic material (in a device) permits electrical and optical signals to “talk” to each other through an “easily perturbed” electron distribution in the material. A low frequency (DC to 200 GHz) electric field (e.g., a television [analog] or computer [digital] signal) is used to perturb the electron distribution (e.g., p-electrons of an organic chromophore) and that perturbation alters the speed of light passing through the material as the electric field component of light (photons) interacts with the perturbed charge distribution.

Because the speed of light is altered by the application of a control voltage, electro-optic materials can be described as materials with a voltage-controlled index of refraction.

For example, you apply and electric field that alters the charge distribution of the material, which in turn influences the propagation of light through the material. (Pockels effect). The reverse process is called optical rectification. When there are two fields involved this is called a second-order nonlinear optical effect.

<div id="Flash">Electro Optic Effect Animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/eo_lightspeed.swf</swf>

In this Flash animation a light source emits photons which travel through the material at the speed of light. When there is no field the electro-optic material has no induced electron asymmetry. Click the battery to add an electric field. The EO materials change their electron distribution which changes their index of refraction so as to slow down light moving through the eo polymer. If these two light beams recombined their wave behavior might interfere. It is this property that can be used to modulate light.

=== Types of EO materials. ===

The response speed of EO materials relates to the mass of the entity that is moved.

'''Liquid Crystals''' -In liquid crystalline materials there is a change in molecular orientation, which changes the dipole moment and charge distribution of the material, which is turn changes the velocity of light moving through the material. This can be measured by the retardation of the speed of light measure in picometers per volt applied. This is a large effect (>10,000 picometers (pm)/V) but rather slow (103 -106 sec) because we are moving a lot of mass. This not so useful for high speed communication.

'''Inorganic crystals''' the electric field causes ion diplacement. This is a small effect (30pm/V) but faster (10-10 sec) because a smaller ion with less mass is moving.

'''electron chromophore polymer'''- A third technique uses π electron chromophore containing polymers and dendrimers. Electric field can change their π electron distribution. This has a large EO activity (>500 pm/v) and very fast into the terahertz (thz) region (10-14 sec).
 
[[Image:organic_modulation_speed.png|thumb|400px|The advantage of organic molecules is high frequency modulation.]]

Organic EO materials have the potential for faster response, lower drive voltage, larger bandwidth, lighter weight and lower cost. They can also be tailored to specific applications and integrated at the chip scale level.
 

== Polarization Effects ==
=== NLO Chromophore ===

[[Image:PASchromophore.JPG|thumb|300px|]]
The basic unit of organic electro-optics is the EO-active material, or chromophore.

This chromophore can be thought of as a molecular oscillator interacting with EM radiation.

Electron donor and acceptor moeties are connected by a π -conjugated bridge that serves as a conduit for electron density.

 
=== Asymmetric Polarization ===
[[Image:4-nitroaniline.png|thumb|300px|4-nitroaniline]]

In second order non linear optics we are concerned with asymmetric polarization of light absorbing molecules in a material.

[[Image:Nlo_effect.png|thumb|500px|Linear and nonlinear polarization response to electric field]]

The diagram is a representation of what happens to a molecule that is asymmetric when an electric field is applied. A molecule with a dipole such as 4-nitroaniline has a charge distribution that leads to a dipole. One side is a donor (d) and an acceptor (a) with a π conjugated system. The magnitude of the induced dipole will be greatest when the electric field is aligned so as to move the electron density towards the electron donor end of the molecule. In a symmetric molecule is there a linear polarizability shown as the straight line. The greater the charge, the greater the induced dipole. In an asymmetric material there a nonlinear effect which makes it easier to polarize in one direction than the other, and increasing electric field has an exponentially increasing effect.

In the presence of an oscillating electric field a linear material will have an induced dipole that is in phase and has the same frequency as the applied field.
 
[[Image:Polarizationwave.png|thumb|300px|An asymmetric polarization response to a symmetric oscillating field]]

The application of a symmetric field (i.e. the electric field associated with the light wave) to the electrons in an anharmonic potential leads to an asymmetric polarization response. This polarization wave has flatted troughs (diminished maxima) in one direction and sharper and higher peaks (accentuated maxima) in the opposite direction, with respect to a normal sine wave.

It is possible to find the sum of waves that would result in such a wave using techniques such as fourier transform. In the case of a symmetric polarization it is simply the sine wave of the applied field.

This asymmetric polarization can be Fourier decomposed (deconvolved) into a static DC polarization component with components at the fundamental frequency superimposed with a second harmonic frequency (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluorescence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/07 Assymetric Polarization.swf</swf>

=== Fourier Analysis of Asymmetric Polarization Wave ===
[[Image:Fourier_harmonics.png|thumb|300px|Combining a fundamental wave and a second harmonic to get a complex polarization wave]]
This asymmetric polarization can be Fourier decomposed (deconvoluted) into a static DC polarization component and components at the fundamental frequency superimposed with a second harmonic frequencies (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluoresence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

=== Expression for Microscopic Nonlinear Polarizabilities ===

The linear polarizability is the first derivative of the dipole moment with respect to electric field.
In non-linear optical effects the plot of induced polarization vs applied field can be corrected using higher corrections with a Taylor series expansion, including the second derivative of the dipole moment with respect to electric field times the field squared with a single electric field, or higher order terms using the third derivative of dipole moment vs field the field cubed. Μ is the total dipole moment in the molecule which is a sum of the static dipole plus several field dependent term.

:<math>\mu = \mu_0 + (\partial \mu_i / \partial E_j)_{E_0}E_j \quad + \quad 1/2 (\partial^2 \mu_i / \partial E_jE_k)_{E_0} E_jE_k \quad+ \quad 1/6(\partial^3\mu_i / \partial E_jE_kE_j)_{E_0} E_jE_kE_j\,\!</math>

Where
:<math>\mu\,\!</math> is the microscopic nonlinear polarizability
:<math>E_i E_k E_j\,\!</math> are the electric field (vectors)
:<math>\mu = \mu_0 \quad+\quad \alpha_{ij}E_j \quad+\quad \beta _{ijk}/ 2 E E \quad+\quad \gamma_{ijkl} / 6 E E E + ...\,\!</math>

Where:

:<math>\alpha\,\!</math> is linear polarizability
:<math>\beta\,\!</math> is the [[first hyperpolarizability]] ( a third rank tensor with 27 permutations although some are degenerate)
:<math>\gamma\,\!</math> is the [[second hyperpolarizability]], responsible for third order non linear optics.

The terms beyond αE are not linear (they have exponential terms) in E and are therefore referred to as the nonlinear polarization and give rise to nonlinear optical effects. Note that Ej, Ek, and Ej are vectors representing the direction of the polarization of the applied field with respect to the molecular coordinate frame. Molecules are asymmetric have different polarizabilities depending the direction of the applied electric field.

Alpha is the second derivative of the dipole moment with respect to field, and is also the first derivative of the polarizability with respect to field. Beta is the first derivative of polarizability with respect to field, and gamma is the first derivative of the first hyperpolarizability with respect to field.

Nonlinear polarization becomes more important with increasing field strength, since it scales with higher powers of the field (quadratic or cubic relationships). Second harmonic generation was not observed until 1961 after the advent of the laser. Under normal conditions,
:<math>\alpha_{ij}E \quad > \quad \beta_{ijk}/2 E·E \quad > \quad \gamma_{ijkl} /6 E·E·E.\,\!</math>

Thus, there were few observations of NLO effects with normal light before the invention of the laser with its associated large electric fields.

With very large electric fields there can be dielectric breakdown of the material.

The observed bulk polarization density is given by an
expression analogous to (7):

:<math>P = P_o + \chi^{(1)} ·E + \chi^{(2)}·· EE + \chi^{(3)}···EEE+ ...\,\!</math> (8)

where the :<math>\chi^{(i)}\,\!</math> susceptibility coefficients are tensors of order i+1 (e.g., :<math>\chi^{(2)}_{ijk}\,\!</math>).
Po is the intrinsic static dipole moment density of the sample.

The linear polarizability is the ability to polarize a molecule, the linear susceptibility is bulk polarization density in a materials which has to do with the polarizability of the molecules and the density of those molecules in the material. More molecules means a higher susceptibility.

=== Taylor Expansion for Bulk Polarization ===
Consider a simple molecule with all the fields being identical.

:<math>P = P_o + \chi^{(1)} ·E + 1/2\chi^{(2)}·· E^2 + 1/6\chi^{(3)}···E^3+ ...\,\!</math>

In a Taylor series expansion the dots refer the fact that these are tensor products. Just as a molecule can only have a non-zero beta if it is noncentrosymmetric, a material can only have a :<math>\chi^{(2)}\,\!</math> if the material is noncentrosymmetric (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules lead to zero :<math>\chi^{(2)}\,\!</math>) .

In a centrosymmetric material a perturbation by an electric field (E) leads to a polarization P. Therefore, application of an electric field (–E) must lead to a polarization –P.

Now consider the second order polarization in a centrosymmetric material.

:<math>P = \chi^{(2)}·· E^2,\,\!</math> (10)

:<math> –P = \chi^{(2)}·· (–E)^2 = \chi^{(2)}·· E^2\,\!</math> (11)

This only occurs when P = 0, therefore :<math>\chi^{(2)}\,\!</math> must be 0.

This means that if we use quantum mechanics to design molecules that will have large hyperpolarizabilities the effort will be wasted if the molecules arrange themselves in a centrosymmetric manner resulting in bulk susceptibility of zero. The design therefore must include both arranging for the desired electronic properties, but also configuring the molecule so that those molecules will not line up in a centrosymmetric manner in the material. A solution of molecules can also exhibit some centrosymmetry.

== Frequency Effects ==

=== Frequency Doubling and Sum-Frequency Generation ===

One nonlinear optical phenomena is that when you shine light at one frequency on a material you get out light with twice the frequency. This process is known as second harmonic generation (SHG). SHG is a special type of sum frequency generation (SFG). SFG occurs when ω3 = ω1 + ω2. If ω3 = ω1 – ω2, this results in difference frequency generation (DFG). These processes are described in more detail below.

The electronic charge displacement (polarization) induced by an oscillating electric field (e.g., light) can be viewed as a classical oscillating dipole that itself emits radiation at the oscillation frequency.

For linear first-order polarization, the radiation has the same frequency as the incident light.

=== Taylor Expansion with Oscillating Electric Fields-SHG ===

The electric field of a plane light wave can be expressed as

:<math>E = E_0 cos(\omega t)\,\!</math>

Using a power series expansion :<math>Ecos^2(\omega t) E\,\!</math> can be substituted for E

:<math>P = P_0 + \chi^{(1)}E_0 cos(\omega t) + \chi^{(2)} E_0^2cos^2(\omega t) + \chi^{(3)} E_0^3 cos^3(\omega t) + ...\,\!</math>

Where
:<math>P_0\,\!</math> is the static polarizablity

Since
:<math>cos^2(\omega t)\,\!</math> equals :<math>1/2 + 1/2 cos(2 \omega t)\,\!</math>,

the first three terms of equation (13) become:

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (14)

This is the origin of the process of optical rectification and second harmonic generation.

=== Second Harmonic Generation (SHG) ===

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (16)

Physically, equation (16) states that the polarization consists of a:

*Second-order DC field contribution to the static polarization (first term),

*Frequency component ω corresponding to the light at the incident frequency (second term) and

*A new frequency doubled component, :<math>2\omega\,\!</math> (third term)-- recall the asymmetric polarization wave and its Fourier analysis.

=== Sum and Difference Frequency Generation ===

In the more general case (in which the two fields are not constrained to be equal), NLO effects involves the interaction of NLO material with two distinct waves with electric fields E1 with the electrons of the NLO material.

Consider two laser beams E1 and E2, the second-order term of equation (4) becomes:
:<math>\chi^{(2)}·E_1cos(\omega_1t)E_2cos(\omega_2t)\,\!</math> (15)

From trigonometry we know that equation (15) is equivalent to:
:<math>1/2\chi^{(2)}·E_1E_2cos [(\omega_1 + \omega_2)t] +1/2\chi^{(2)}·E_1E_2cos [(\omega_1 - \omega_2)t]\,\!</math> (16)

Thus when two light beams of frequencies ω1 and ω2 interact in an NLO material, polarization occurs at sum :<math>(\omega_1 + \omega_2)\,\!</math> and difference :<math>(\omega_1 - \omega_2)\,\!</math> frequencies.

This electronic polarization will therefore, re-emit radiation at these frequencies.

The combination of frequencies is called sum (or difference) frequency generation (SFG) of which SHG is a special case. This is how a tunable laser works.

Note that a very short laser pulse will result in a band or distribution of frequencies due to the Heisenberg Uncertainty Principle. Those bands will add and subtract resulting in some light which is twice the frequency if they added, and some light that is very low frequency (0+ or – the difference), resulting from the difference between the frequencies. This is the process enabling Terahertz spectroscopy. Terahertz is very low frequency light.

Low frequency light is scattered less than high frequency light. For example if you look through a glass of milk there is “index inhomogeneity” in the milk due the presence of protein and fat. Terahertz radiation can be used for surveillance. A terahertz detector scanner will reveal materials that have different index of refraction.

== Electro-optic effects ==

=== Kerr and Pockels Effects ===

John Kerr and Friedrich Pockels discovered in 1875 and 1893, respectively, that the refractive index of a material could be changed by applying a DC or low frequency electric field. This are in fact non-linear optical effects but they often not thought of as such because they don’t require a laser.

Electric impermeability of a material can be expressed as:

:<math>n \equiv \frac {\epsilon_0}{\epsilon} = \frac{1}{n^2}\,\!</math>

:<math>\eta (E) = \eta + rE +SE^2\,\!</math>

Where
:<math>\epsilon_0\,\!</math> is the dielectric constant of free space
:<math>\epsilon\,\!</math> is the dielectric constant

'''Pockels effect'''
[[Image:Pockels_graph.png|thumb|200px|The Pockels effect has a linear relation to applied field]]
In the Pockels effect an applied electric field changes the refractive index of certain materials.

:<math>\eta(E) = \eta - \frac {1} {2}rn^3 E\,\!</math>

Where:

'''r''' is Pockels coefficient or Linear Electro-optic Coefficient, r~ 10-12 – 1—10 m/V, typically.

This is a linear function with respect to the electric field, the higher the r the greater the change. It is cubic with respect to the refractive index so materials with high intrinsic refractive indexes will change more. Some examples include NH4H2PO4(ADP), KH2PO4(KDP), LiNbO3, LiTaO3, CdTe
 
'''The Kerr effect'''
[[Image:Kerr_graph.png|thumb|200px|The Kerr effect has a parabolic relationship to applied field]]
:<math>\eta(E) = \eta – ½ Sn^3E^2\,\!</math>

Where

'''S''' is the Kerr coefficient
*S~ 10-18 – 10-14 mV in crystals
*S~ 10-22 – 10-19 mV in liquids

This is similar to the Pockels effect except that the refractive index varies parabolically or quadratically with the electric field.

This a process that occurs in second order nonlinear optical materials. It is a third order nonlinear optical process. Not all materials are second order nonlinear optical materials, only those that are centrosymmetric. However all materials have a χ(3) even if they are centrosymmetric.

It is possible to change the amplitude, phase or path of light at a given frequency by using a static DC electric field to polarize the material and modify the refractive indices. When light enters a material with a higher refractive index it is phase shifted and the waves become compressed. The direction is also changed. So by changing the refractive index it is possible to change the path of the light.

Consider the special case :<math>\omega_2 = 0\,\!</math> [equation (15)] in which a DC electric field is applied to the material.

The optical frequency polarization (Popt) arising from the second-order susceptibility is given by:

:<math>P_{opt} =\chi^{(2)}·E_1E_2(cos \omega_1 t)\,\!</math> (17)

where:
E1
E2 is the magnitude of the electric field caused by voltage applied to the nonlinear material (a voltage not optical frequency).

Recall that the refractive index is related to the linear susceptibility that is given by the second term of Equation (14):

:<math>\chi^{(1)}·E_1(cos \omega_1 t)\,\!</math> (18)

The total optical frequency polarization (Popt) is the χ(1) term plus the χ(2) term:

:<math>P_{opt} = \chi^{(1)}·E_1(cos_1t) +\chi^{(2)}·E_1E_2(cos \omega_1t)\,\!</math> (19)

Then factor out <math>E_1(cos \omega_1t)\,\!</math> :

:<math> P_{opt} = [\chi^{(1)} + \chi^{(2)}·E_2] E_1(cos \omega_1t)\,\!</math> (20)

χ(1) is linear susceptability which relates to the dielectric constant, which in turn relates to the square of the refractive index. A change in the linear susceptablity changes the index of refraction. The second term: χ(2) times the magnitude of the voltage (E2) means that the susceptability of the material, the dielectric constant of the material, and the refractive index of the material can be altered by changing the applied voltage.

You can shine light on second order nonlinear optical materials and get out different frequencies, or shine one laser beam, apply an electric field and then modulate the refractive index. For example, light can travel freely between two fibers that are very close to each other with the same refractive index. But if the fibers have a different refractive index light will stay in one fiber or the other.

By changing the refractive index you can move light from one fiber to another; it provides a means of switching light in waveguides.

'''Summary'''

*The applied field in effect changes the linear susceptibility and thus the refractive index of the material.

*This is, known as the linear electro-optic (LEO) or Pockels effect, and is used to modulate light by changing the applied voltage.

*At the atomic level, the applied voltage is anisotropically distorting the electron density within the material. Thus, application of a voltage to the material causes the optical beam to "see" a different material with a different polarizability and a different anisotropy of the polarizability than in the absence of the voltage.

*Since the anisotropy is changed upon application of an electric field, a beam of light can have its polarization state (i.e., ellipticity) changed by an amount related to the strength and orientation of the applied voltage, and travel at a different speed and possibly in a different direction.

=== Index modulation ===

Quantitatively, the change in the refractive index as a function of the applied electric field is approximated by
the general expression:

:<math>1/\underline{n}_{ij}2 = 1/n_{ij}2 + r_{ijk}E_k + s_{ijkl}E_kE_l + ... \,\!</math> (21)

where
:<math>\underline{n}_{ij}\,\!</math> are the induced refractive indices,
:<math>n_{ij}\,\!</math> is the refractive index in the absence of the electric field,

:<math>r_{ijk}\,\!</math> is the linear or Pockels coefficients, Δn for E = 106 V/m is 10-6 to 10-4 (crystals) and;

:<math>s_{ijkl}\,\!</math> are the quadratic or Kerr coefficients.

=== r coefficients ===

The optical indicatrix (that characterizes the anisotropy of the refractive index) therefore changes as the electric field within the sample changes. The of map the index of refraction with respect to each polarization of light produces a surface that looks something like a football. The electric field allows you to change the shape of the football.

Electro-optic coefficients are frequently defined in terms of rijk.
The "r" coefficients form a tensor (just as do the coefficient of α).

The subscripts ijk are the same as those used with β. The first subscript (i) refers to the resultant polarization of the material along a defined axis and the following subscripts j and k refer to the orientations of the applied fields, one is the optical frequency field and k is the voltage.

=== Applications of Electro-optic Devices ===
[[Image:Network.png|thumb|400px|EO materials can be used at many locations in a network]]
A network has a variety of devices that provide input from to a transmitter, connected by a electro-optic modulator (EOM) through a switching network, to a receiver with a photodetector, and then are connected to display devices. Nonlinear optical materials can be used for any of these applications. They can used to create terahertz radiation and to create specific wavelengths of light for spectroscopy.
 

[[category:non linear optics]]
[[category:second order NLO]]
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Second-order Processes

2011-08-12T23:30:25Z

Chrisb: /* Taylor Expansion with Oscillating Electric Fields-SHG */

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[http://depts.washington.edu/cmditr/media/NLO_materials.html Concept Map for Second Order Non linear Optics]

Second order non linear optics involve the search for materials whose optical properties can be controlled with an applied electrical or optical field. The are second order because the effect is quadratic with respect to field strength. These extremely fast processes can be used for optical switching in telecommunication and the frequency effects can be used for specialized spectroscopy, imaging and scanning.

== Electro optical materials ==

=== EO Materials have a voltage-controlled index of refraction. ===
Light has a known speed in a vacuum. But when enters a material it slows down. Light has a electrical and magnetic component. The electrical component will interact with the charge distribution of the atom in the material is passed through. The interaction will slow the light down.

The index of refraction = speed of light in vacuum / speed of light in material.

An electro-optic material (in a device) permits electrical and optical signals to “talk” to each other through an “easily perturbed” electron distribution in the material. A low frequency (DC to 200 GHz) electric field (e.g., a television [analog] or computer [digital] signal) is used to perturb the electron distribution (e.g., p-electrons of an organic chromophore) and that perturbation alters the speed of light passing through the material as the electric field component of light (photons) interacts with the perturbed charge distribution.

Because the speed of light is altered by the application of a control voltage, electro-optic materials can be described as materials with a voltage-controlled index of refraction.

For example, you apply and electric field that alters the charge distribution of the material, which in turn influences the propagation of light through the material. (Pockels effect). The reverse process is called optical rectification. When there are two fields involved this is called a second-order nonlinear optical effect.

<div id="Flash">Electro Optic Effect Animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/eo_lightspeed.swf</swf>

In this Flash animation a light source emits photons which travel through the material at the speed of light. When there is no field the electro-optic material has no induced electron asymmetry. Click the battery to add an electric field. The EO materials change their electron distribution which changes their index of refraction so as to slow down light moving through the eo polymer. If these two light beams recombined their wave behavior might interfere. It is this property that can be used to modulate light.

=== Types of EO materials. ===

The response speed of EO materials relates to the mass of the entity that is moved.

'''Liquid Crystals''' -In liquid crystalline materials there is a change in molecular orientation, which changes the dipole moment and charge distribution of the material, which is turn changes the velocity of light moving through the material. This can be measured by the retardation of the speed of light measure in picometers per volt applied. This is a large effect (>10,000 picometers (pm)/V) but rather slow (103 -106 sec) because we are moving a lot of mass. This not so useful for high speed communication.

'''Inorganic crystals''' the electric field causes ion diplacement. This is a small effect (30pm/V) but faster (10-10 sec) because a smaller ion with less mass is moving.

'''electron chromophore polymer'''- A third technique uses π electron chromophore containing polymers and dendrimers. Electric field can change their π electron distribution. This has a large EO activity (>500 pm/v) and very fast into the terahertz (thz) region (10-14 sec).
 
[[Image:organic_modulation_speed.png|thumb|400px|The advantage of organic molecules is high frequency modulation.]]

Organic EO materials have the potential for faster response, lower drive voltage, larger bandwidth, lighter weight and lower cost. They can also be tailored to specific applications and integrated at the chip scale level.
 

== Polarization Effects ==
=== NLO Chromophore ===

[[Image:PASchromophore.JPG|thumb|300px|]]
The basic unit of organic electro-optics is the EO-active material, or chromophore.

This chromophore can be thought of as a molecular oscillator interacting with EM radiation.

Electron donor and acceptor moeties are connected by a π -conjugated bridge that serves as a conduit for electron density.

 
=== Asymmetric Polarization ===
[[Image:4-nitroaniline.png|thumb|300px|4-nitroaniline]]

In second order non linear optics we are concerned with asymmetric polarization of light absorbing molecules in a material.

[[Image:Nlo_effect.png|thumb|500px|Linear and nonlinear polarization response to electric field]]

The diagram is a representation of what happens to a molecule that is asymmetric when an electric field is applied. A molecule with a dipole such as 4-nitroaniline has a charge distribution that leads to a dipole. One side is a donor (d) and an acceptor (a) with a π conjugated system. The magnitude of the induced dipole will be greatest when the electric field is aligned so as to move the electron density towards the electron donor end of the molecule. In a symmetric molecule is there a linear polarizability shown as the straight line. The greater the charge, the greater the induced dipole. In an asymmetric material there a nonlinear effect which makes it easier to polarize in one direction than the other, and increasing electric field has an exponentially increasing effect.

In the presence of an oscillating electric field a linear material will have an induced dipole that is in phase and has the same frequency as the applied field.
 
[[Image:Polarizationwave.png|thumb|300px|An asymmetric polarization response to a symmetric oscillating field]]

The application of a symmetric field (i.e. the electric field associated with the light wave) to the electrons in an anharmonic potential leads to an asymmetric polarization response. This polarization wave has flatted troughs (diminished maxima) in one direction and sharper and higher peaks (accentuated maxima) in the opposite direction, with respect to a normal sine wave.

It is possible to find the sum of waves that would result in such a wave using techniques such as fourier transform. In the case of a symmetric polarization it is simply the sine wave of the applied field.

This asymmetric polarization can be Fourier decomposed (deconvolved) into a static DC polarization component with components at the fundamental frequency superimposed with a second harmonic frequency (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluorescence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/07 Assymetric Polarization.swf</swf>

=== Fourier Analysis of Asymmetric Polarization Wave ===
[[Image:Fourier_harmonics.png|thumb|300px|Combining a fundamental wave and a second harmonic to get a complex polarization wave]]
This asymmetric polarization can be Fourier decomposed (deconvoluted) into a static DC polarization component and components at the fundamental frequency superimposed with a second harmonic frequencies (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluoresence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

=== Expression for Microscopic Nonlinear Polarizabilities ===

The linear polarizability is the first derivative of the dipole moment with respect to electric field.
In non-linear optical effects the plot of induced polarization vs applied field can be corrected using higher corrections with a Taylor series expansion, including the second derivative of the dipole moment with respect to electric field times the field squared with a single electric field, or higher order terms using the third derivative of dipole moment vs field the field cubed. Μ is the total dipole moment in the molecule which is a sum of the static dipole plus several field dependent term.

:<math>\mu = \mu_0 + (\partial \mu_i / \partial E_j)_{E_0}E_j \quad + \quad 1/2 (\partial^2 \mu_i / \partial E_jE_k)_{E_0} E_jE_k \quad+ \quad 1/6(\partial^3\mu_i / \partial E_jE_kE_j)_{E_0} E_jE_kE_j\,\!</math>

Where
:<math>\mu\,\!</math> is the microscopic nonlinear polarizability
:<math>E_i E_k E_j\,\!</math> are the electric field (vectors)
:<math>\mu = \mu_0 \quad+\quad \alpha_{ij}E_j \quad+\quad \beta _{ijk}/ 2 E E \quad+\quad \gamma_{ijkl} / 6 E E E + ...\,\!</math>

Where:

:<math>\alpha\,\!</math> is linear polarizability
:<math>\beta\,\!</math> is the [[first hyperpolarizability]] ( a third rank tensor with 27 permutations although some are degenerate)
:<math>\gamma\,\!</math> is the [[second hyperpolarizability]], responsible for third order non linear optics.

The terms beyond αE are not linear (they have exponential terms) in E and are therefore referred to as the nonlinear polarization and give rise to nonlinear optical effects. Note that Ej, Ek, and Ej are vectors representing the direction of the polarization of the applied field with respect to the molecular coordinate frame. Molecules are asymmetric have different polarizabilities depending the direction of the applied electric field.

Alpha is the second derivative of the dipole moment with respect to field, and is also the first derivative of the polarizability with respect to field. Beta is the first derivative of polarizability with respect to field, and gamma is the first derivative of the first hyperpolarizability with respect to field.

Nonlinear polarization becomes more important with increasing field strength, since it scales with higher powers of the field (quadratic or cubic relationships). Second harmonic generation was not observed until 1961 after the advent of the laser. Under normal conditions,
:<math>\alpha_{ij}E \quad > \quad \beta_{ijk}/2 E·E \quad > \quad \gamma_{ijkl} /6 E·E·E.\,\!</math>

Thus, there were few observations of NLO effects with normal light before the invention of the laser with its associated large electric fields.

With very large electric fields there can be dielectric breakdown of the material.

The observed bulk polarization density is given by an
expression analogous to (7):

:<math>P = P_o + \chi^{(1)} ·E + \chi^{(2)}·· EE + \chi^{(3)}···EEE+ ...\,\!</math> (8)

where the :<math>\chi^{(i)}\,\!</math> susceptibility coefficients are tensors of order i+1 (e.g., :<math>\chi^{(2)}_{ijk}\,\!</math>).
Po is the intrinsic static dipole moment density of the sample.

The linear polarizability is the ability to polarize a molecule, the linear susceptibility is bulk polarization density in a materials which has to do with the polarizability of the molecules and the density of those molecules in the material. More molecules means a higher susceptibility.

=== Taylor Expansion for Bulk Polarization ===
Consider a simple molecule with all the fields being identical.

:<math>P = P_o + \chi^{(1)} ·E + 1/2\chi^{(2)}·· E^2 + 1/6\chi^{(3)}···E^3+ ...\,\!</math>

In a Taylor series expansion the dots refer the fact that these are tensor products. Just as a molecule can only have a non-zero beta if it is noncentrosymmetric, a material can only have a :<math>\chi^{(2)}\,\!</math> if the material is noncentrosymmetric (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules lead to zero :<math>\chi^{(2)}\,\!</math>) .

In a centrosymmetric material a perturbation by an electric field (E) leads to a polarization P. Therefore, application of an electric field (–E) must lead to a polarization –P.

Now consider the second order polarization in a centrosymmetric material.

:<math>P = \chi^{(2)}·· E^2,\,\!</math> (10)

:<math> –P = \chi^{(2)}·· (–E)^2 = \chi^{(2)}·· E^2\,\!</math> (11)

This only occurs when P = 0, therefore :<math>\chi^{(2)}\,\!</math> must be 0.

This means that if we use quantum mechanics to design molecules that will have large hyperpolarizabilities the effort will be wasted if the molecules arrange themselves in a centrosymmetric manner resulting in bulk susceptibility of zero. The design therefore must include both arranging for the desired electronic properties, but also configuring the molecule so that those molecules will not line up in a centrosymmetric manner in the material. A solution of molecules can also exhibit some centrosymmetry.

== Frequency Effects ==

=== Frequency Doubling and Sum-Frequency Generation ===

One nonlinear optical phenomena is that when you shine light at one frequency on a material you get out light with twice the frequency. This process is known as second harmonic generation (SHG). SHG is a special type of sum frequency generation (SFG). SFG occurs when ω3 = ω1 + ω2. If ω3 = ω1 – ω2, this results in difference frequency generation (DFG). These processes are used to generate different frequencies of light, and thus new wavelengths of light can be used in experiments.

The electronic charge displacement (polarization) induced by an oscillating electric field (e.g., light) can be viewed as a classical oscillating dipole that itself emits radiation at the oscillation frequency.

For linear first-order polarization, the radiation has the same frequency as the incident light.

=== Taylor Expansion with Oscillating Electric Fields-SHG ===

The electric field of a plane light wave can be expressed as

:<math>E = E_0 cos(\omega t)\,\!</math>

Using a power series expansion :<math>Ecos^2(\omega t) E\,\!</math> can be substituted for E

:<math>P = P_0 + \chi^{(1)}E_0 cos(\omega t) + \chi^{(2)} E_0^2cos^2(\omega t) + \chi^{(3)} E_0^3 cos^3(\omega t) + ...\,\!</math>

Where
:<math>P_0\,\!</math> is the static polarizablity

Since
:<math>cos^2(\omega t)\,\!</math> equals :<math>1/2 + 1/2 cos(2 \omega t)\,\!</math>,

the first three terms of equation (13) become:

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (14)

This is the origin of the process of optical rectification and second harmonic generation.

=== Second Harmonic Generation (SHG) ===

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (16)

Physically, equation (16) states that the polarization consists of a:

*Second-order DC field contribution to the static polarization (first term),

*Frequency component ω corresponding to the light at the incident frequency (second term) and

*A new frequency doubled component, :<math>2\omega\,\!</math> (third term)-- recall the asymmetric polarization wave and its Fourier analysis.

=== Sum and Difference Frequency Generation ===

In the more general case (in which the two fields are not constrained to be equal), NLO effects involves the interaction of NLO material with two distinct waves with electric fields E1 with the electrons of the NLO material.

Consider two laser beams E1 and E2, the second-order term of equation (4) becomes:
:<math>\chi^{(2)}·E_1cos(\omega_1t)E_2cos(\omega_2t)\,\!</math> (15)

From trigonometry we know that equation (15) is equivalent to:
:<math>1/2\chi^{(2)}·E_1E_2cos [(\omega_1 + \omega_2)t] +1/2\chi^{(2)}·E_1E_2cos [(\omega_1 - \omega_2)t]\,\!</math> (16)

Thus when two light beams of frequencies ω1 and ω2 interact in an NLO material, polarization occurs at sum :<math>(\omega_1 + \omega_2)\,\!</math> and difference :<math>(\omega_1 - \omega_2)\,\!</math> frequencies.

This electronic polarization will therefore, re-emit radiation at these frequencies.

The combination of frequencies is called sum (or difference) frequency generation (SFG) of which SHG is a special case. This is how a tunable laser works.

Note that a very short laser pulse will result in a band or distribution of frequencies due to the Heisenberg Uncertainty Principle. Those bands will add and subtract resulting in some light which is twice the frequency if they added, and some light that is very low frequency (0+ or – the difference), resulting from the difference between the frequencies. This is the process enabling Terahertz spectroscopy. Terahertz is very low frequency light.

Low frequency light is scattered less than high frequency light. For example if you look through a glass of milk there is “index inhomogeneity” in the milk due the presence of protein and fat. Terahertz radiation can be used for surveillance. A terahertz detector scanner will reveal materials that have different index of refraction.

== Electro-optic effects ==

=== Kerr and Pockels Effects ===

John Kerr and Friedrich Pockels discovered in 1875 and 1893, respectively, that the refractive index of a material could be changed by applying a DC or low frequency electric field. This are in fact non-linear optical effects but they often not thought of as such because they don’t require a laser.

Electric impermeability of a material can be expressed as:

:<math>n \equiv \frac {\epsilon_0}{\epsilon} = \frac{1}{n^2}\,\!</math>

:<math>\eta (E) = \eta + rE +SE^2\,\!</math>

Where
:<math>\epsilon_0\,\!</math> is the dielectric constant of free space
:<math>\epsilon\,\!</math> is the dielectric constant

'''Pockels effect'''
[[Image:Pockels_graph.png|thumb|200px|The Pockels effect has a linear relation to applied field]]
In the Pockels effect an applied electric field changes the refractive index of certain materials.

:<math>\eta(E) = \eta - \frac {1} {2}rn^3 E\,\!</math>

Where:

'''r''' is Pockels coefficient or Linear Electro-optic Coefficient, r~ 10-12 – 1—10 m/V, typically.

This is a linear function with respect to the electric field, the higher the r the greater the change. It is cubic with respect to the refractive index so materials with high intrinsic refractive indexes will change more. Some examples include NH4H2PO4(ADP), KH2PO4(KDP), LiNbO3, LiTaO3, CdTe
 
'''The Kerr effect'''
[[Image:Kerr_graph.png|thumb|200px|The Kerr effect has a parabolic relationship to applied field]]
:<math>\eta(E) = \eta – ½ Sn^3E^2\,\!</math>

Where

'''S''' is the Kerr coefficient
*S~ 10-18 – 10-14 mV in crystals
*S~ 10-22 – 10-19 mV in liquids

This is similar to the Pockels effect except that the refractive index varies parabolically or quadratically with the electric field.

This a process that occurs in second order nonlinear optical materials. It is a third order nonlinear optical process. Not all materials are second order nonlinear optical materials, only those that are centrosymmetric. However all materials have a χ(3) even if they are centrosymmetric.

It is possible to change the amplitude, phase or path of light at a given frequency by using a static DC electric field to polarize the material and modify the refractive indices. When light enters a material with a higher refractive index it is phase shifted and the waves become compressed. The direction is also changed. So by changing the refractive index it is possible to change the path of the light.

Consider the special case :<math>\omega_2 = 0\,\!</math> [equation (15)] in which a DC electric field is applied to the material.

The optical frequency polarization (Popt) arising from the second-order susceptibility is given by:

:<math>P_{opt} =\chi^{(2)}·E_1E_2(cos \omega_1 t)\,\!</math> (17)

where:
E1
E2 is the magnitude of the electric field caused by voltage applied to the nonlinear material (a voltage not optical frequency).

Recall that the refractive index is related to the linear susceptibility that is given by the second term of Equation (14):

:<math>\chi^{(1)}·E_1(cos \omega_1 t)\,\!</math> (18)

The total optical frequency polarization (Popt) is the χ(1) term plus the χ(2) term:

:<math>P_{opt} = \chi^{(1)}·E_1(cos_1t) +\chi^{(2)}·E_1E_2(cos \omega_1t)\,\!</math> (19)

Then factor out <math>E_1(cos \omega_1t)\,\!</math> :

:<math> P_{opt} = [\chi^{(1)} + \chi^{(2)}·E_2] E_1(cos \omega_1t)\,\!</math> (20)

χ(1) is linear susceptability which relates to the dielectric constant, which in turn relates to the square of the refractive index. A change in the linear susceptablity changes the index of refraction. The second term: χ(2) times the magnitude of the voltage (E2) means that the susceptability of the material, the dielectric constant of the material, and the refractive index of the material can be altered by changing the applied voltage.

You can shine light on second order nonlinear optical materials and get out different frequencies, or shine one laser beam, apply an electric field and then modulate the refractive index. For example, light can travel freely between two fibers that are very close to each other with the same refractive index. But if the fibers have a different refractive index light will stay in one fiber or the other.

By changing the refractive index you can move light from one fiber to another; it provides a means of switching light in waveguides.

'''Summary'''

*The applied field in effect changes the linear susceptibility and thus the refractive index of the material.

*This is, known as the linear electro-optic (LEO) or Pockels effect, and is used to modulate light by changing the applied voltage.

*At the atomic level, the applied voltage is anisotropically distorting the electron density within the material. Thus, application of a voltage to the material causes the optical beam to "see" a different material with a different polarizability and a different anisotropy of the polarizability than in the absence of the voltage.

*Since the anisotropy is changed upon application of an electric field, a beam of light can have its polarization state (i.e., ellipticity) changed by an amount related to the strength and orientation of the applied voltage, and travel at a different speed and possibly in a different direction.

=== Index modulation ===

Quantitatively, the change in the refractive index as a function of the applied electric field is approximated by
the general expression:

:<math>1/\underline{n}_{ij}2 = 1/n_{ij}2 + r_{ijk}E_k + s_{ijkl}E_kE_l + ... \,\!</math> (21)

where
:<math>\underline{n}_{ij}\,\!</math> are the induced refractive indices,
:<math>n_{ij}\,\!</math> is the refractive index in the absence of the electric field,

:<math>r_{ijk}\,\!</math> is the linear or Pockels coefficients, Δn for E = 106 V/m is 10-6 to 10-4 (crystals) and;

:<math>s_{ijkl}\,\!</math> are the quadratic or Kerr coefficients.

=== r coefficients ===

The optical indicatrix (that characterizes the anisotropy of the refractive index) therefore changes as the electric field within the sample changes. The of map the index of refraction with respect to each polarization of light produces a surface that looks something like a football. The electric field allows you to change the shape of the football.

Electro-optic coefficients are frequently defined in terms of rijk.
The "r" coefficients form a tensor (just as do the coefficient of α).

The subscripts ijk are the same as those used with β. The first subscript (i) refers to the resultant polarization of the material along a defined axis and the following subscripts j and k refer to the orientations of the applied fields, one is the optical frequency field and k is the voltage.

=== Applications of Electro-optic Devices ===
[[Image:Network.png|thumb|400px|EO materials can be used at many locations in a network]]
A network has a variety of devices that provide input from to a transmitter, connected by a electro-optic modulator (EOM) through a switching network, to a receiver with a photodetector, and then are connected to display devices. Nonlinear optical materials can be used for any of these applications. They can used to create terahertz radiation and to create specific wavelengths of light for spectroscopy.
 

[[category:non linear optics]]
[[category:second order NLO]]
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Second-order Processes

2011-08-12T23:25:26Z

Chrisb: /* Frequency Doubling and Sum-Frequency Generation */

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[http://depts.washington.edu/cmditr/media/NLO_materials.html Concept Map for Second Order Non linear Optics]

Second order non linear optics involve the search for materials whose optical properties can be controlled with an applied electrical or optical field. The are second order because the effect is quadratic with respect to field strength. These extremely fast processes can be used for optical switching in telecommunication and the frequency effects can be used for specialized spectroscopy, imaging and scanning.

== Electro optical materials ==

=== EO Materials have a voltage-controlled index of refraction. ===
Light has a known speed in a vacuum. But when enters a material it slows down. Light has a electrical and magnetic component. The electrical component will interact with the charge distribution of the atom in the material is passed through. The interaction will slow the light down.

The index of refraction = speed of light in vacuum / speed of light in material.

An electro-optic material (in a device) permits electrical and optical signals to “talk” to each other through an “easily perturbed” electron distribution in the material. A low frequency (DC to 200 GHz) electric field (e.g., a television [analog] or computer [digital] signal) is used to perturb the electron distribution (e.g., p-electrons of an organic chromophore) and that perturbation alters the speed of light passing through the material as the electric field component of light (photons) interacts with the perturbed charge distribution.

Because the speed of light is altered by the application of a control voltage, electro-optic materials can be described as materials with a voltage-controlled index of refraction.

For example, you apply and electric field that alters the charge distribution of the material, which in turn influences the propagation of light through the material. (Pockels effect). The reverse process is called optical rectification. When there are two fields involved this is called a second-order nonlinear optical effect.

<div id="Flash">Electro Optic Effect Animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/eo_lightspeed.swf</swf>

In this Flash animation a light source emits photons which travel through the material at the speed of light. When there is no field the electro-optic material has no induced electron asymmetry. Click the battery to add an electric field. The EO materials change their electron distribution which changes their index of refraction so as to slow down light moving through the eo polymer. If these two light beams recombined their wave behavior might interfere. It is this property that can be used to modulate light.

=== Types of EO materials. ===

The response speed of EO materials relates to the mass of the entity that is moved.

'''Liquid Crystals''' -In liquid crystalline materials there is a change in molecular orientation, which changes the dipole moment and charge distribution of the material, which is turn changes the velocity of light moving through the material. This can be measured by the retardation of the speed of light measure in picometers per volt applied. This is a large effect (>10,000 picometers (pm)/V) but rather slow (103 -106 sec) because we are moving a lot of mass. This not so useful for high speed communication.

'''Inorganic crystals''' the electric field causes ion diplacement. This is a small effect (30pm/V) but faster (10-10 sec) because a smaller ion with less mass is moving.

'''electron chromophore polymer'''- A third technique uses π electron chromophore containing polymers and dendrimers. Electric field can change their π electron distribution. This has a large EO activity (>500 pm/v) and very fast into the terahertz (thz) region (10-14 sec).
 
[[Image:organic_modulation_speed.png|thumb|400px|The advantage of organic molecules is high frequency modulation.]]

Organic EO materials have the potential for faster response, lower drive voltage, larger bandwidth, lighter weight and lower cost. They can also be tailored to specific applications and integrated at the chip scale level.
 

== Polarization Effects ==
=== NLO Chromophore ===

[[Image:PASchromophore.JPG|thumb|300px|]]
The basic unit of organic electro-optics is the EO-active material, or chromophore.

This chromophore can be thought of as a molecular oscillator interacting with EM radiation.

Electron donor and acceptor moeties are connected by a π -conjugated bridge that serves as a conduit for electron density.

 
=== Asymmetric Polarization ===
[[Image:4-nitroaniline.png|thumb|300px|4-nitroaniline]]

In second order non linear optics we are concerned with asymmetric polarization of light absorbing molecules in a material.

[[Image:Nlo_effect.png|thumb|500px|Linear and nonlinear polarization response to electric field]]

The diagram is a representation of what happens to a molecule that is asymmetric when an electric field is applied. A molecule with a dipole such as 4-nitroaniline has a charge distribution that leads to a dipole. One side is a donor (d) and an acceptor (a) with a π conjugated system. The magnitude of the induced dipole will be greatest when the electric field is aligned so as to move the electron density towards the electron donor end of the molecule. In a symmetric molecule is there a linear polarizability shown as the straight line. The greater the charge, the greater the induced dipole. In an asymmetric material there a nonlinear effect which makes it easier to polarize in one direction than the other, and increasing electric field has an exponentially increasing effect.

In the presence of an oscillating electric field a linear material will have an induced dipole that is in phase and has the same frequency as the applied field.
 
[[Image:Polarizationwave.png|thumb|300px|An asymmetric polarization response to a symmetric oscillating field]]

The application of a symmetric field (i.e. the electric field associated with the light wave) to the electrons in an anharmonic potential leads to an asymmetric polarization response. This polarization wave has flatted troughs (diminished maxima) in one direction and sharper and higher peaks (accentuated maxima) in the opposite direction, with respect to a normal sine wave.

It is possible to find the sum of waves that would result in such a wave using techniques such as fourier transform. In the case of a symmetric polarization it is simply the sine wave of the applied field.

This asymmetric polarization can be Fourier decomposed (deconvolved) into a static DC polarization component with components at the fundamental frequency superimposed with a second harmonic frequency (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluorescence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/07 Assymetric Polarization.swf</swf>

=== Fourier Analysis of Asymmetric Polarization Wave ===
[[Image:Fourier_harmonics.png|thumb|300px|Combining a fundamental wave and a second harmonic to get a complex polarization wave]]
This asymmetric polarization can be Fourier decomposed (deconvoluted) into a static DC polarization component and components at the fundamental frequency superimposed with a second harmonic frequencies (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluoresence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

=== Expression for Microscopic Nonlinear Polarizabilities ===

The linear polarizability is the first derivative of the dipole moment with respect to electric field.
In non-linear optical effects the plot of induced polarization vs applied field can be corrected using higher corrections with a Taylor series expansion, including the second derivative of the dipole moment with respect to electric field times the field squared with a single electric field, or higher order terms using the third derivative of dipole moment vs field the field cubed. Μ is the total dipole moment in the molecule which is a sum of the static dipole plus several field dependent term.

:<math>\mu = \mu_0 + (\partial \mu_i / \partial E_j)_{E_0}E_j \quad + \quad 1/2 (\partial^2 \mu_i / \partial E_jE_k)_{E_0} E_jE_k \quad+ \quad 1/6(\partial^3\mu_i / \partial E_jE_kE_j)_{E_0} E_jE_kE_j\,\!</math>

Where
:<math>\mu\,\!</math> is the microscopic nonlinear polarizability
:<math>E_i E_k E_j\,\!</math> are the electric field (vectors)
:<math>\mu = \mu_0 \quad+\quad \alpha_{ij}E_j \quad+\quad \beta _{ijk}/ 2 E E \quad+\quad \gamma_{ijkl} / 6 E E E + ...\,\!</math>

Where:

:<math>\alpha\,\!</math> is linear polarizability
:<math>\beta\,\!</math> is the [[first hyperpolarizability]] ( a third rank tensor with 27 permutations although some are degenerate)
:<math>\gamma\,\!</math> is the [[second hyperpolarizability]], responsible for third order non linear optics.

The terms beyond αE are not linear (they have exponential terms) in E and are therefore referred to as the nonlinear polarization and give rise to nonlinear optical effects. Note that Ej, Ek, and Ej are vectors representing the direction of the polarization of the applied field with respect to the molecular coordinate frame. Molecules are asymmetric have different polarizabilities depending the direction of the applied electric field.

Alpha is the second derivative of the dipole moment with respect to field, and is also the first derivative of the polarizability with respect to field. Beta is the first derivative of polarizability with respect to field, and gamma is the first derivative of the first hyperpolarizability with respect to field.

Nonlinear polarization becomes more important with increasing field strength, since it scales with higher powers of the field (quadratic or cubic relationships). Second harmonic generation was not observed until 1961 after the advent of the laser. Under normal conditions,
:<math>\alpha_{ij}E \quad > \quad \beta_{ijk}/2 E·E \quad > \quad \gamma_{ijkl} /6 E·E·E.\,\!</math>

Thus, there were few observations of NLO effects with normal light before the invention of the laser with its associated large electric fields.

With very large electric fields there can be dielectric breakdown of the material.

The observed bulk polarization density is given by an
expression analogous to (7):

:<math>P = P_o + \chi^{(1)} ·E + \chi^{(2)}·· EE + \chi^{(3)}···EEE+ ...\,\!</math> (8)

where the :<math>\chi^{(i)}\,\!</math> susceptibility coefficients are tensors of order i+1 (e.g., :<math>\chi^{(2)}_{ijk}\,\!</math>).
Po is the intrinsic static dipole moment density of the sample.

The linear polarizability is the ability to polarize a molecule, the linear susceptibility is bulk polarization density in a materials which has to do with the polarizability of the molecules and the density of those molecules in the material. More molecules means a higher susceptibility.

=== Taylor Expansion for Bulk Polarization ===
Consider a simple molecule with all the fields being identical.

:<math>P = P_o + \chi^{(1)} ·E + 1/2\chi^{(2)}·· E^2 + 1/6\chi^{(3)}···E^3+ ...\,\!</math>

In a Taylor series expansion the dots refer the fact that these are tensor products. Just as a molecule can only have a non-zero beta if it is noncentrosymmetric, a material can only have a :<math>\chi^{(2)}\,\!</math> if the material is noncentrosymmetric (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules lead to zero :<math>\chi^{(2)}\,\!</math>) .

In a centrosymmetric material a perturbation by an electric field (E) leads to a polarization P. Therefore, application of an electric field (–E) must lead to a polarization –P.

Now consider the second order polarization in a centrosymmetric material.

:<math>P = \chi^{(2)}·· E^2,\,\!</math> (10)

:<math> –P = \chi^{(2)}·· (–E)^2 = \chi^{(2)}·· E^2\,\!</math> (11)

This only occurs when P = 0, therefore :<math>\chi^{(2)}\,\!</math> must be 0.

This means that if we use quantum mechanics to design molecules that will have large hyperpolarizabilities the effort will be wasted if the molecules arrange themselves in a centrosymmetric manner resulting in bulk susceptibility of zero. The design therefore must include both arranging for the desired electronic properties, but also configuring the molecule so that those molecules will not line up in a centrosymmetric manner in the material. A solution of molecules can also exhibit some centrosymmetry.

== Frequency Effects ==

=== Frequency Doubling and Sum-Frequency Generation ===

One nonlinear optical phenomena is that when you shine light at one frequency on a material you get out light with twice the frequency. This process is known as second harmonic generation (SHG). SHG is a special type of sum frequency generation (SFG). SFG occurs when ω3 = ω1 + ω2. If ω3 = ω1 – ω2, this results in difference frequency generation (DFG). These processes are used to generate different frequencies of light, and thus new wavelengths of light can be used in experiments.

The electronic charge displacement (polarization) induced by an oscillating electric field (e.g., light) can be viewed as a classical oscillating dipole that itself emits radiation at the oscillation frequency.

For linear first-order polarization, the radiation has the same frequency as the incident light.

=== Taylor Expansion with Oscillating Electric Fields-SHG ===

The electric field of a plane light wave can be expressed as

:<math>E = E_0 cos(\omega t)\,\!</math>

Using a power series expansion :<math>Ecos^2(\omega t) E\,\!</math> can be substituted for E

:<math>P = (P_0 + \chi^{(1)}E_0 cos(\omega t) + \chi^{(2)} E_0^2cos^2(\omega t) + \chi^{(3)} E_0^3 cos^3(\omega t) + ...\,\!</math>

Where
:<math>P_0\,\!</math> is the static polarizablity

Since
:<math>cos^2(\omega t)\,\!</math> equals :<math>1/2 + 1/2 cos(2 \omega t)\,\!</math>,

the first three terms of equation (13) become:

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (14)

This is the origin of the process of optical rectification and second harmonic generation.

=== Second Harmonic Generation (SHG) ===

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (16)

Physically, equation (16) states that the polarization consists of a:

*Second-order DC field contribution to the static polarization (first term),

*Frequency component ω corresponding to the light at the incident frequency (second term) and

*A new frequency doubled component, :<math>2\omega\,\!</math> (third term)-- recall the asymmetric polarization wave and its Fourier analysis.

=== Sum and Difference Frequency Generation ===

In the more general case (in which the two fields are not constrained to be equal), NLO effects involves the interaction of NLO material with two distinct waves with electric fields E1 with the electrons of the NLO material.

Consider two laser beams E1 and E2, the second-order term of equation (4) becomes:
:<math>\chi^{(2)}·E_1cos(\omega_1t)E_2cos(\omega_2t)\,\!</math> (15)

From trigonometry we know that equation (15) is equivalent to:
:<math>1/2\chi^{(2)}·E_1E_2cos [(\omega_1 + \omega_2)t] +1/2\chi^{(2)}·E_1E_2cos [(\omega_1 - \omega_2)t]\,\!</math> (16)

Thus when two light beams of frequencies ω1 and ω2 interact in an NLO material, polarization occurs at sum :<math>(\omega_1 + \omega_2)\,\!</math> and difference :<math>(\omega_1 - \omega_2)\,\!</math> frequencies.

This electronic polarization will therefore, re-emit radiation at these frequencies.

The combination of frequencies is called sum (or difference) frequency generation (SFG) of which SHG is a special case. This is how a tunable laser works.

Note that a very short laser pulse will result in a band or distribution of frequencies due to the Heisenberg Uncertainty Principle. Those bands will add and subtract resulting in some light which is twice the frequency if they added, and some light that is very low frequency (0+ or – the difference), resulting from the difference between the frequencies. This is the process enabling Terahertz spectroscopy. Terahertz is very low frequency light.

Low frequency light is scattered less than high frequency light. For example if you look through a glass of milk there is “index inhomogeneity” in the milk due the presence of protein and fat. Terahertz radiation can be used for surveillance. A terahertz detector scanner will reveal materials that have different index of refraction.

== Electro-optic effects ==

=== Kerr and Pockels Effects ===

John Kerr and Friedrich Pockels discovered in 1875 and 1893, respectively, that the refractive index of a material could be changed by applying a DC or low frequency electric field. This are in fact non-linear optical effects but they often not thought of as such because they don’t require a laser.

Electric impermeability of a material can be expressed as:

:<math>n \equiv \frac {\epsilon_0}{\epsilon} = \frac{1}{n^2}\,\!</math>

:<math>\eta (E) = \eta + rE +SE^2\,\!</math>

Where
:<math>\epsilon_0\,\!</math> is the dielectric constant of free space
:<math>\epsilon\,\!</math> is the dielectric constant

'''Pockels effect'''
[[Image:Pockels_graph.png|thumb|200px|The Pockels effect has a linear relation to applied field]]
In the Pockels effect an applied electric field changes the refractive index of certain materials.

:<math>\eta(E) = \eta - \frac {1} {2}rn^3 E\,\!</math>

Where:

'''r''' is Pockels coefficient or Linear Electro-optic Coefficient, r~ 10-12 – 1—10 m/V, typically.

This is a linear function with respect to the electric field, the higher the r the greater the change. It is cubic with respect to the refractive index so materials with high intrinsic refractive indexes will change more. Some examples include NH4H2PO4(ADP), KH2PO4(KDP), LiNbO3, LiTaO3, CdTe
 
'''The Kerr effect'''
[[Image:Kerr_graph.png|thumb|200px|The Kerr effect has a parabolic relationship to applied field]]
:<math>\eta(E) = \eta – ½ Sn^3E^2\,\!</math>

Where

'''S''' is the Kerr coefficient
*S~ 10-18 – 10-14 mV in crystals
*S~ 10-22 – 10-19 mV in liquids

This is similar to the Pockels effect except that the refractive index varies parabolically or quadratically with the electric field.

This a process that occurs in second order nonlinear optical materials. It is a third order nonlinear optical process. Not all materials are second order nonlinear optical materials, only those that are centrosymmetric. However all materials have a χ(3) even if they are centrosymmetric.

It is possible to change the amplitude, phase or path of light at a given frequency by using a static DC electric field to polarize the material and modify the refractive indices. When light enters a material with a higher refractive index it is phase shifted and the waves become compressed. The direction is also changed. So by changing the refractive index it is possible to change the path of the light.

Consider the special case :<math>\omega_2 = 0\,\!</math> [equation (15)] in which a DC electric field is applied to the material.

The optical frequency polarization (Popt) arising from the second-order susceptibility is given by:

:<math>P_{opt} =\chi^{(2)}·E_1E_2(cos \omega_1 t)\,\!</math> (17)

where:
E1
E2 is the magnitude of the electric field caused by voltage applied to the nonlinear material (a voltage not optical frequency).

Recall that the refractive index is related to the linear susceptibility that is given by the second term of Equation (14):

:<math>\chi^{(1)}·E_1(cos \omega_1 t)\,\!</math> (18)

The total optical frequency polarization (Popt) is the χ(1) term plus the χ(2) term:

:<math>P_{opt} = \chi^{(1)}·E_1(cos_1t) +\chi^{(2)}·E_1E_2(cos \omega_1t)\,\!</math> (19)

Then factor out <math>E_1(cos \omega_1t)\,\!</math> :

:<math> P_{opt} = [\chi^{(1)} + \chi^{(2)}·E_2] E_1(cos \omega_1t)\,\!</math> (20)

χ(1) is linear susceptability which relates to the dielectric constant, which in turn relates to the square of the refractive index. A change in the linear susceptablity changes the index of refraction. The second term: χ(2) times the magnitude of the voltage (E2) means that the susceptability of the material, the dielectric constant of the material, and the refractive index of the material can be altered by changing the applied voltage.

You can shine light on second order nonlinear optical materials and get out different frequencies, or shine one laser beam, apply an electric field and then modulate the refractive index. For example, light can travel freely between two fibers that are very close to each other with the same refractive index. But if the fibers have a different refractive index light will stay in one fiber or the other.

By changing the refractive index you can move light from one fiber to another; it provides a means of switching light in waveguides.

'''Summary'''

*The applied field in effect changes the linear susceptibility and thus the refractive index of the material.

*This is, known as the linear electro-optic (LEO) or Pockels effect, and is used to modulate light by changing the applied voltage.

*At the atomic level, the applied voltage is anisotropically distorting the electron density within the material. Thus, application of a voltage to the material causes the optical beam to "see" a different material with a different polarizability and a different anisotropy of the polarizability than in the absence of the voltage.

*Since the anisotropy is changed upon application of an electric field, a beam of light can have its polarization state (i.e., ellipticity) changed by an amount related to the strength and orientation of the applied voltage, and travel at a different speed and possibly in a different direction.

=== Index modulation ===

Quantitatively, the change in the refractive index as a function of the applied electric field is approximated by
the general expression:

:<math>1/\underline{n}_{ij}2 = 1/n_{ij}2 + r_{ijk}E_k + s_{ijkl}E_kE_l + ... \,\!</math> (21)

where
:<math>\underline{n}_{ij}\,\!</math> are the induced refractive indices,
:<math>n_{ij}\,\!</math> is the refractive index in the absence of the electric field,

:<math>r_{ijk}\,\!</math> is the linear or Pockels coefficients, Δn for E = 106 V/m is 10-6 to 10-4 (crystals) and;

:<math>s_{ijkl}\,\!</math> are the quadratic or Kerr coefficients.

=== r coefficients ===

The optical indicatrix (that characterizes the anisotropy of the refractive index) therefore changes as the electric field within the sample changes. The of map the index of refraction with respect to each polarization of light produces a surface that looks something like a football. The electric field allows you to change the shape of the football.

Electro-optic coefficients are frequently defined in terms of rijk.
The "r" coefficients form a tensor (just as do the coefficient of α).

The subscripts ijk are the same as those used with β. The first subscript (i) refers to the resultant polarization of the material along a defined axis and the following subscripts j and k refer to the orientations of the applied fields, one is the optical frequency field and k is the voltage.

=== Applications of Electro-optic Devices ===
[[Image:Network.png|thumb|400px|EO materials can be used at many locations in a network]]
A network has a variety of devices that provide input from to a transmitter, connected by a electro-optic modulator (EOM) through a switching network, to a receiver with a photodetector, and then are connected to display devices. Nonlinear optical materials can be used for any of these applications. They can used to create terahertz radiation and to create specific wavelengths of light for spectroscopy.
 

[[category:non linear optics]]
[[category:second order NLO]]
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Second-order Processes

2011-08-12T23:25:02Z

Chrisb: /* Frequency Doubling and Sum-Frequency Generation */

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[http://depts.washington.edu/cmditr/media/NLO_materials.html Concept Map for Second Order Non linear Optics]

Second order non linear optics involve the search for materials whose optical properties can be controlled with an applied electrical or optical field. The are second order because the effect is quadratic with respect to field strength. These extremely fast processes can be used for optical switching in telecommunication and the frequency effects can be used for specialized spectroscopy, imaging and scanning.

== Electro optical materials ==

=== EO Materials have a voltage-controlled index of refraction. ===
Light has a known speed in a vacuum. But when enters a material it slows down. Light has a electrical and magnetic component. The electrical component will interact with the charge distribution of the atom in the material is passed through. The interaction will slow the light down.

The index of refraction = speed of light in vacuum / speed of light in material.

An electro-optic material (in a device) permits electrical and optical signals to “talk” to each other through an “easily perturbed” electron distribution in the material. A low frequency (DC to 200 GHz) electric field (e.g., a television [analog] or computer [digital] signal) is used to perturb the electron distribution (e.g., p-electrons of an organic chromophore) and that perturbation alters the speed of light passing through the material as the electric field component of light (photons) interacts with the perturbed charge distribution.

Because the speed of light is altered by the application of a control voltage, electro-optic materials can be described as materials with a voltage-controlled index of refraction.

For example, you apply and electric field that alters the charge distribution of the material, which in turn influences the propagation of light through the material. (Pockels effect). The reverse process is called optical rectification. When there are two fields involved this is called a second-order nonlinear optical effect.

<div id="Flash">Electro Optic Effect Animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/eo_lightspeed.swf</swf>

In this Flash animation a light source emits photons which travel through the material at the speed of light. When there is no field the electro-optic material has no induced electron asymmetry. Click the battery to add an electric field. The EO materials change their electron distribution which changes their index of refraction so as to slow down light moving through the eo polymer. If these two light beams recombined their wave behavior might interfere. It is this property that can be used to modulate light.

=== Types of EO materials. ===

The response speed of EO materials relates to the mass of the entity that is moved.

'''Liquid Crystals''' -In liquid crystalline materials there is a change in molecular orientation, which changes the dipole moment and charge distribution of the material, which is turn changes the velocity of light moving through the material. This can be measured by the retardation of the speed of light measure in picometers per volt applied. This is a large effect (>10,000 picometers (pm)/V) but rather slow (103 -106 sec) because we are moving a lot of mass. This not so useful for high speed communication.

'''Inorganic crystals''' the electric field causes ion diplacement. This is a small effect (30pm/V) but faster (10-10 sec) because a smaller ion with less mass is moving.

'''electron chromophore polymer'''- A third technique uses π electron chromophore containing polymers and dendrimers. Electric field can change their π electron distribution. This has a large EO activity (>500 pm/v) and very fast into the terahertz (thz) region (10-14 sec).
 
[[Image:organic_modulation_speed.png|thumb|400px|The advantage of organic molecules is high frequency modulation.]]

Organic EO materials have the potential for faster response, lower drive voltage, larger bandwidth, lighter weight and lower cost. They can also be tailored to specific applications and integrated at the chip scale level.
 

== Polarization Effects ==
=== NLO Chromophore ===

[[Image:PASchromophore.JPG|thumb|300px|]]
The basic unit of organic electro-optics is the EO-active material, or chromophore.

This chromophore can be thought of as a molecular oscillator interacting with EM radiation.

Electron donor and acceptor moeties are connected by a π -conjugated bridge that serves as a conduit for electron density.

 
=== Asymmetric Polarization ===
[[Image:4-nitroaniline.png|thumb|300px|4-nitroaniline]]

In second order non linear optics we are concerned with asymmetric polarization of light absorbing molecules in a material.

[[Image:Nlo_effect.png|thumb|500px|Linear and nonlinear polarization response to electric field]]

The diagram is a representation of what happens to a molecule that is asymmetric when an electric field is applied. A molecule with a dipole such as 4-nitroaniline has a charge distribution that leads to a dipole. One side is a donor (d) and an acceptor (a) with a π conjugated system. The magnitude of the induced dipole will be greatest when the electric field is aligned so as to move the electron density towards the electron donor end of the molecule. In a symmetric molecule is there a linear polarizability shown as the straight line. The greater the charge, the greater the induced dipole. In an asymmetric material there a nonlinear effect which makes it easier to polarize in one direction than the other, and increasing electric field has an exponentially increasing effect.

In the presence of an oscillating electric field a linear material will have an induced dipole that is in phase and has the same frequency as the applied field.
 
[[Image:Polarizationwave.png|thumb|300px|An asymmetric polarization response to a symmetric oscillating field]]

The application of a symmetric field (i.e. the electric field associated with the light wave) to the electrons in an anharmonic potential leads to an asymmetric polarization response. This polarization wave has flatted troughs (diminished maxima) in one direction and sharper and higher peaks (accentuated maxima) in the opposite direction, with respect to a normal sine wave.

It is possible to find the sum of waves that would result in such a wave using techniques such as fourier transform. In the case of a symmetric polarization it is simply the sine wave of the applied field.

This asymmetric polarization can be Fourier decomposed (deconvolved) into a static DC polarization component with components at the fundamental frequency superimposed with a second harmonic frequency (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluorescence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/07 Assymetric Polarization.swf</swf>

=== Fourier Analysis of Asymmetric Polarization Wave ===
[[Image:Fourier_harmonics.png|thumb|300px|Combining a fundamental wave and a second harmonic to get a complex polarization wave]]
This asymmetric polarization can be Fourier decomposed (deconvoluted) into a static DC polarization component and components at the fundamental frequency superimposed with a second harmonic frequencies (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluoresence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

=== Expression for Microscopic Nonlinear Polarizabilities ===

The linear polarizability is the first derivative of the dipole moment with respect to electric field.
In non-linear optical effects the plot of induced polarization vs applied field can be corrected using higher corrections with a Taylor series expansion, including the second derivative of the dipole moment with respect to electric field times the field squared with a single electric field, or higher order terms using the third derivative of dipole moment vs field the field cubed. Μ is the total dipole moment in the molecule which is a sum of the static dipole plus several field dependent term.

:<math>\mu = \mu_0 + (\partial \mu_i / \partial E_j)_{E_0}E_j \quad + \quad 1/2 (\partial^2 \mu_i / \partial E_jE_k)_{E_0} E_jE_k \quad+ \quad 1/6(\partial^3\mu_i / \partial E_jE_kE_j)_{E_0} E_jE_kE_j\,\!</math>

Where
:<math>\mu\,\!</math> is the microscopic nonlinear polarizability
:<math>E_i E_k E_j\,\!</math> are the electric field (vectors)
:<math>\mu = \mu_0 \quad+\quad \alpha_{ij}E_j \quad+\quad \beta _{ijk}/ 2 E E \quad+\quad \gamma_{ijkl} / 6 E E E + ...\,\!</math>

Where:

:<math>\alpha\,\!</math> is linear polarizability
:<math>\beta\,\!</math> is the [[first hyperpolarizability]] ( a third rank tensor with 27 permutations although some are degenerate)
:<math>\gamma\,\!</math> is the [[second hyperpolarizability]], responsible for third order non linear optics.

The terms beyond αE are not linear (they have exponential terms) in E and are therefore referred to as the nonlinear polarization and give rise to nonlinear optical effects. Note that Ej, Ek, and Ej are vectors representing the direction of the polarization of the applied field with respect to the molecular coordinate frame. Molecules are asymmetric have different polarizabilities depending the direction of the applied electric field.

Alpha is the second derivative of the dipole moment with respect to field, and is also the first derivative of the polarizability with respect to field. Beta is the first derivative of polarizability with respect to field, and gamma is the first derivative of the first hyperpolarizability with respect to field.

Nonlinear polarization becomes more important with increasing field strength, since it scales with higher powers of the field (quadratic or cubic relationships). Second harmonic generation was not observed until 1961 after the advent of the laser. Under normal conditions,
:<math>\alpha_{ij}E \quad > \quad \beta_{ijk}/2 E·E \quad > \quad \gamma_{ijkl} /6 E·E·E.\,\!</math>

Thus, there were few observations of NLO effects with normal light before the invention of the laser with its associated large electric fields.

With very large electric fields there can be dielectric breakdown of the material.

The observed bulk polarization density is given by an
expression analogous to (7):

:<math>P = P_o + \chi^{(1)} ·E + \chi^{(2)}·· EE + \chi^{(3)}···EEE+ ...\,\!</math> (8)

where the :<math>\chi^{(i)}\,\!</math> susceptibility coefficients are tensors of order i+1 (e.g., :<math>\chi^{(2)}_{ijk}\,\!</math>).
Po is the intrinsic static dipole moment density of the sample.

The linear polarizability is the ability to polarize a molecule, the linear susceptibility is bulk polarization density in a materials which has to do with the polarizability of the molecules and the density of those molecules in the material. More molecules means a higher susceptibility.

=== Taylor Expansion for Bulk Polarization ===
Consider a simple molecule with all the fields being identical.

:<math>P = P_o + \chi^{(1)} ·E + 1/2\chi^{(2)}·· E^2 + 1/6\chi^{(3)}···E^3+ ...\,\!</math>

In a Taylor series expansion the dots refer the fact that these are tensor products. Just as a molecule can only have a non-zero beta if it is noncentrosymmetric, a material can only have a :<math>\chi^{(2)}\,\!</math> if the material is noncentrosymmetric (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules lead to zero :<math>\chi^{(2)}\,\!</math>) .

In a centrosymmetric material a perturbation by an electric field (E) leads to a polarization P. Therefore, application of an electric field (–E) must lead to a polarization –P.

Now consider the second order polarization in a centrosymmetric material.

:<math>P = \chi^{(2)}·· E^2,\,\!</math> (10)

:<math> –P = \chi^{(2)}·· (–E)^2 = \chi^{(2)}·· E^2\,\!</math> (11)

This only occurs when P = 0, therefore :<math>\chi^{(2)}\,\!</math> must be 0.

This means that if we use quantum mechanics to design molecules that will have large hyperpolarizabilities the effort will be wasted if the molecules arrange themselves in a centrosymmetric manner resulting in bulk susceptibility of zero. The design therefore must include both arranging for the desired electronic properties, but also configuring the molecule so that those molecules will not line up in a centrosymmetric manner in the material. A solution of molecules can also exhibit some centrosymmetry.

== Frequency Effects ==

=== Frequency Doubling and Sum-Frequency Generation ===

One nonlinear optical phenomena is that when you shine light at one frequency on a material you get out light with twice the frequency. This process is known as second harmonic generation (SHG). SHG is a special type of sum frequency generation (SFG). SFG occurs when ω3 = ω1 and ω2. If ω3 = ω1 – ω2, this results in difference frequency generation (DFG). These processes are used to generate different frequencies of light, and thus new wavelengths of light can be used in experiments.

The electronic charge displacement (polarization) induced by an oscillating electric field (e.g., light) can be viewed as a classical oscillating dipole that itself emits radiation at the oscillation frequency.

For linear first-order polarization, the radiation has the same frequency as the incident light.

=== Taylor Expansion with Oscillating Electric Fields-SHG ===

The electric field of a plane light wave can be expressed as

:<math>E = E_0 cos(\omega t)\,\!</math>

Using a power series expansion :<math>Ecos^2(\omega t) E\,\!</math> can be substituted for E

:<math>P = (P_0 + \chi^{(1)}E_0 cos(\omega t) + \chi^{(2)} E_0^2cos^2(\omega t) + \chi^{(3)} E_0^3 cos^3(\omega t) + ...\,\!</math>

Where
:<math>P_0\,\!</math> is the static polarizablity

Since
:<math>cos^2(\omega t)\,\!</math> equals :<math>1/2 + 1/2 cos(2 \omega t)\,\!</math>,

the first three terms of equation (13) become:

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (14)

This is the origin of the process of optical rectification and second harmonic generation.

=== Second Harmonic Generation (SHG) ===

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (16)

Physically, equation (16) states that the polarization consists of a:

*Second-order DC field contribution to the static polarization (first term),

*Frequency component ω corresponding to the light at the incident frequency (second term) and

*A new frequency doubled component, :<math>2\omega\,\!</math> (third term)-- recall the asymmetric polarization wave and its Fourier analysis.

=== Sum and Difference Frequency Generation ===

In the more general case (in which the two fields are not constrained to be equal), NLO effects involves the interaction of NLO material with two distinct waves with electric fields E1 with the electrons of the NLO material.

Consider two laser beams E1 and E2, the second-order term of equation (4) becomes:
:<math>\chi^{(2)}·E_1cos(\omega_1t)E_2cos(\omega_2t)\,\!</math> (15)

From trigonometry we know that equation (15) is equivalent to:
:<math>1/2\chi^{(2)}·E_1E_2cos [(\omega_1 + \omega_2)t] +1/2\chi^{(2)}·E_1E_2cos [(\omega_1 - \omega_2)t]\,\!</math> (16)

Thus when two light beams of frequencies ω1 and ω2 interact in an NLO material, polarization occurs at sum :<math>(\omega_1 + \omega_2)\,\!</math> and difference :<math>(\omega_1 - \omega_2)\,\!</math> frequencies.

This electronic polarization will therefore, re-emit radiation at these frequencies.

The combination of frequencies is called sum (or difference) frequency generation (SFG) of which SHG is a special case. This is how a tunable laser works.

Note that a very short laser pulse will result in a band or distribution of frequencies due to the Heisenberg Uncertainty Principle. Those bands will add and subtract resulting in some light which is twice the frequency if they added, and some light that is very low frequency (0+ or – the difference), resulting from the difference between the frequencies. This is the process enabling Terahertz spectroscopy. Terahertz is very low frequency light.

Low frequency light is scattered less than high frequency light. For example if you look through a glass of milk there is “index inhomogeneity” in the milk due the presence of protein and fat. Terahertz radiation can be used for surveillance. A terahertz detector scanner will reveal materials that have different index of refraction.

== Electro-optic effects ==

=== Kerr and Pockels Effects ===

John Kerr and Friedrich Pockels discovered in 1875 and 1893, respectively, that the refractive index of a material could be changed by applying a DC or low frequency electric field. This are in fact non-linear optical effects but they often not thought of as such because they don’t require a laser.

Electric impermeability of a material can be expressed as:

:<math>n \equiv \frac {\epsilon_0}{\epsilon} = \frac{1}{n^2}\,\!</math>

:<math>\eta (E) = \eta + rE +SE^2\,\!</math>

Where
:<math>\epsilon_0\,\!</math> is the dielectric constant of free space
:<math>\epsilon\,\!</math> is the dielectric constant

'''Pockels effect'''
[[Image:Pockels_graph.png|thumb|200px|The Pockels effect has a linear relation to applied field]]
In the Pockels effect an applied electric field changes the refractive index of certain materials.

:<math>\eta(E) = \eta - \frac {1} {2}rn^3 E\,\!</math>

Where:

'''r''' is Pockels coefficient or Linear Electro-optic Coefficient, r~ 10-12 – 1—10 m/V, typically.

This is a linear function with respect to the electric field, the higher the r the greater the change. It is cubic with respect to the refractive index so materials with high intrinsic refractive indexes will change more. Some examples include NH4H2PO4(ADP), KH2PO4(KDP), LiNbO3, LiTaO3, CdTe
 
'''The Kerr effect'''
[[Image:Kerr_graph.png|thumb|200px|The Kerr effect has a parabolic relationship to applied field]]
:<math>\eta(E) = \eta – ½ Sn^3E^2\,\!</math>

Where

'''S''' is the Kerr coefficient
*S~ 10-18 – 10-14 mV in crystals
*S~ 10-22 – 10-19 mV in liquids

This is similar to the Pockels effect except that the refractive index varies parabolically or quadratically with the electric field.

This a process that occurs in second order nonlinear optical materials. It is a third order nonlinear optical process. Not all materials are second order nonlinear optical materials, only those that are centrosymmetric. However all materials have a χ(3) even if they are centrosymmetric.

It is possible to change the amplitude, phase or path of light at a given frequency by using a static DC electric field to polarize the material and modify the refractive indices. When light enters a material with a higher refractive index it is phase shifted and the waves become compressed. The direction is also changed. So by changing the refractive index it is possible to change the path of the light.

Consider the special case :<math>\omega_2 = 0\,\!</math> [equation (15)] in which a DC electric field is applied to the material.

The optical frequency polarization (Popt) arising from the second-order susceptibility is given by:

:<math>P_{opt} =\chi^{(2)}·E_1E_2(cos \omega_1 t)\,\!</math> (17)

where:
E1
E2 is the magnitude of the electric field caused by voltage applied to the nonlinear material (a voltage not optical frequency).

Recall that the refractive index is related to the linear susceptibility that is given by the second term of Equation (14):

:<math>\chi^{(1)}·E_1(cos \omega_1 t)\,\!</math> (18)

The total optical frequency polarization (Popt) is the χ(1) term plus the χ(2) term:

:<math>P_{opt} = \chi^{(1)}·E_1(cos_1t) +\chi^{(2)}·E_1E_2(cos \omega_1t)\,\!</math> (19)

Then factor out <math>E_1(cos \omega_1t)\,\!</math> :

:<math> P_{opt} = [\chi^{(1)} + \chi^{(2)}·E_2] E_1(cos \omega_1t)\,\!</math> (20)

χ(1) is linear susceptability which relates to the dielectric constant, which in turn relates to the square of the refractive index. A change in the linear susceptablity changes the index of refraction. The second term: χ(2) times the magnitude of the voltage (E2) means that the susceptability of the material, the dielectric constant of the material, and the refractive index of the material can be altered by changing the applied voltage.

You can shine light on second order nonlinear optical materials and get out different frequencies, or shine one laser beam, apply an electric field and then modulate the refractive index. For example, light can travel freely between two fibers that are very close to each other with the same refractive index. But if the fibers have a different refractive index light will stay in one fiber or the other.

By changing the refractive index you can move light from one fiber to another; it provides a means of switching light in waveguides.

'''Summary'''

*The applied field in effect changes the linear susceptibility and thus the refractive index of the material.

*This is, known as the linear electro-optic (LEO) or Pockels effect, and is used to modulate light by changing the applied voltage.

*At the atomic level, the applied voltage is anisotropically distorting the electron density within the material. Thus, application of a voltage to the material causes the optical beam to "see" a different material with a different polarizability and a different anisotropy of the polarizability than in the absence of the voltage.

*Since the anisotropy is changed upon application of an electric field, a beam of light can have its polarization state (i.e., ellipticity) changed by an amount related to the strength and orientation of the applied voltage, and travel at a different speed and possibly in a different direction.

=== Index modulation ===

Quantitatively, the change in the refractive index as a function of the applied electric field is approximated by
the general expression:

:<math>1/\underline{n}_{ij}2 = 1/n_{ij}2 + r_{ijk}E_k + s_{ijkl}E_kE_l + ... \,\!</math> (21)

where
:<math>\underline{n}_{ij}\,\!</math> are the induced refractive indices,
:<math>n_{ij}\,\!</math> is the refractive index in the absence of the electric field,

:<math>r_{ijk}\,\!</math> is the linear or Pockels coefficients, Δn for E = 106 V/m is 10-6 to 10-4 (crystals) and;

:<math>s_{ijkl}\,\!</math> are the quadratic or Kerr coefficients.

=== r coefficients ===

The optical indicatrix (that characterizes the anisotropy of the refractive index) therefore changes as the electric field within the sample changes. The of map the index of refraction with respect to each polarization of light produces a surface that looks something like a football. The electric field allows you to change the shape of the football.

Electro-optic coefficients are frequently defined in terms of rijk.
The "r" coefficients form a tensor (just as do the coefficient of α).

The subscripts ijk are the same as those used with β. The first subscript (i) refers to the resultant polarization of the material along a defined axis and the following subscripts j and k refer to the orientations of the applied fields, one is the optical frequency field and k is the voltage.

=== Applications of Electro-optic Devices ===
[[Image:Network.png|thumb|400px|EO materials can be used at many locations in a network]]
A network has a variety of devices that provide input from to a transmitter, connected by a electro-optic modulator (EOM) through a switching network, to a receiver with a photodetector, and then are connected to display devices. Nonlinear optical materials can be used for any of these applications. They can used to create terahertz radiation and to create specific wavelengths of light for spectroscopy.
 

[[category:non linear optics]]
[[category:second order NLO]]
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Second-order Processes

2011-08-11T17:51:50Z

Chrisb: /* Expression for Microscopic Nonlinear Polarizabilities */

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[http://depts.washington.edu/cmditr/media/NLO_materials.html Concept Map for Second Order Non linear Optics]

Second order non linear optics involve the search for materials whose optical properties can be controlled with an applied electrical or optical field. The are second order because the effect is quadratic with respect to field strength. These extremely fast processes can be used for optical switching in telecommunication and the frequency effects can be used for specialized spectroscopy, imaging and scanning.

== Electro optical materials ==

=== EO Materials have a voltage-controlled index of refraction. ===
Light has a known speed in a vacuum. But when enters a material it slows down. Light has a electrical and magnetic component. The electrical component will interact with the charge distribution of the atom in the material is passed through. The interaction will slow the light down.

The index of refraction = speed of light in vacuum / speed of light in material.

An electro-optic material (in a device) permits electrical and optical signals to “talk” to each other through an “easily perturbed” electron distribution in the material. A low frequency (DC to 200 GHz) electric field (e.g., a television [analog] or computer [digital] signal) is used to perturb the electron distribution (e.g., p-electrons of an organic chromophore) and that perturbation alters the speed of light passing through the material as the electric field component of light (photons) interacts with the perturbed charge distribution.

Because the speed of light is altered by the application of a control voltage, electro-optic materials can be described as materials with a voltage-controlled index of refraction.

For example, you apply and electric field that alters the charge distribution of the material, which in turn influences the propagation of light through the material. (Pockels effect). The reverse process is called optical rectification. When there are two fields involved this is called a second-order nonlinear optical effect.

<div id="Flash">Electro Optic Effect Animation</div>

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/eo_lightspeed.swf</swf>

In this Flash animation a light source emits photons which travel through the material at the speed of light. When there is no field the electro-optic material has no induced electron asymmetry. Click the battery to add an electric field. The EO materials change their electron distribution which changes their index of refraction so as to slow down light moving through the eo polymer. If these two light beams recombined their wave behavior might interfere. It is this property that can be used to modulate light.

=== Types of EO materials. ===

The response speed of EO materials relates to the mass of the entity that is moved.

'''Liquid Crystals''' -In liquid crystalline materials there is a change in molecular orientation, which changes the dipole moment and charge distribution of the material, which is turn changes the velocity of light moving through the material. This can be measured by the retardation of the speed of light measure in picometers per volt applied. This is a large effect (>10,000 picometers (pm)/V) but rather slow (103 -106 sec) because we are moving a lot of mass. This not so useful for high speed communication.

'''Inorganic crystals''' the electric field causes ion diplacement. This is a small effect (30pm/V) but faster (10-10 sec) because a smaller ion with less mass is moving.

'''electron chromophore polymer'''- A third technique uses π electron chromophore containing polymers and dendrimers. Electric field can change their π electron distribution. This has a large EO activity (>500 pm/v) and very fast into the terahertz (thz) region (10-14 sec).
 
[[Image:organic_modulation_speed.png|thumb|400px|The advantage of organic molecules is high frequency modulation.]]

Organic EO materials have the potential for faster response, lower drive voltage, larger bandwidth, lighter weight and lower cost. They can also be tailored to specific applications and integrated at the chip scale level.
 

== Polarization Effects ==
=== NLO Chromophore ===

[[Image:PASchromophore.JPG|thumb|300px|]]
The basic unit of organic electro-optics is the EO-active material, or chromophore.

This chromophore can be thought of as a molecular oscillator interacting with EM radiation.

Electron donor and acceptor moeties are connected by a π -conjugated bridge that serves as a conduit for electron density.

 
=== Asymmetric Polarization ===
[[Image:4-nitroaniline.png|thumb|300px|4-nitroaniline]]

In second order non linear optics we are concerned with asymmetric polarization of light absorbing molecules in a material.

[[Image:Nlo_effect.png|thumb|500px|Linear and nonlinear polarization response to electric field]]

The diagram is a representation of what happens to a molecule that is asymmetric when an electric field is applied. A molecule with a dipole such as 4-nitroaniline has a charge distribution that leads to a dipole. One side is a donor (d) and an acceptor (a) with a π conjugated system. The magnitude of the induced dipole will be greatest when the electric field is aligned so as to move the electron density towards the electron donor end of the molecule. In a symmetric molecule is there a linear polarizability shown as the straight line. The greater the charge, the greater the induced dipole. In an asymmetric material there a nonlinear effect which makes it easier to polarize in one direction than the other, and increasing electric field has an exponentially increasing effect.

In the presence of an oscillating electric field a linear material will have an induced dipole that is in phase and has the same frequency as the applied field.
 
[[Image:Polarizationwave.png|thumb|300px|An asymmetric polarization response to a symmetric oscillating field]]

The application of a symmetric field (i.e. the electric field associated with the light wave) to the electrons in an anharmonic potential leads to an asymmetric polarization response. This polarization wave has flatted troughs (diminished maxima) in one direction and sharper and higher peaks (accentuated maxima) in the opposite direction, with respect to a normal sine wave.

It is possible to find the sum of waves that would result in such a wave using techniques such as fourier transform. In the case of a symmetric polarization it is simply the sine wave of the applied field.

This asymmetric polarization can be Fourier decomposed (deconvolved) into a static DC polarization component with components at the fundamental frequency superimposed with a second harmonic frequency (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluorescence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

<swf width="500" height="400">http://depts.washington.edu/cmditr/media/07 Assymetric Polarization.swf</swf>

=== Fourier Analysis of Asymmetric Polarization Wave ===
[[Image:Fourier_harmonics.png|thumb|300px|Combining a fundamental wave and a second harmonic to get a complex polarization wave]]
This asymmetric polarization can be Fourier decomposed (deconvoluted) into a static DC polarization component and components at the fundamental frequency superimposed with a second harmonic frequencies (at twice the fundamental frequency). This is a consequence of the material having second order non-linear properties.

As a consequence if you shine a laser at a non-linear optical material you can get light out of a different wavelength than the exciting source. In addition this emission can occur at a wavelength where the molecule is completely non-absorbing. This emission is not due to absorption / fluoresence. In second harmonic generation the light coming out can be twice frequency of the exciting source and the phenomena is tunable.

A laser of one frequency can be used to generate light of other frequencies. For example green light from a niodinum YAG laser (1064nm wavelength - green) can be directed on a non-linear optical crystal such as potassium dihydrogen phosphate and generate a second harmonic which is then used as a source for other experiments.

Since only the time averaged asymmetrically induced polarization leads to second-order NLO effects, only molecules and materials lacking a center of symmetry possess them.

=== Expression for Microscopic Nonlinear Polarizabilities ===

The linear polarizability is the first derivative of the dipole moment with respect to electric field.
In non-linear optical effects the plot of induced polarization vs applied field can be corrected using higher corrections with a Taylor series expansion, including the second derivative of the dipole moment with respect to electric field times the field squared with a single electric field, or higher order terms using the third derivative of dipole moment vs field the field cubed. Μ is the total dipole moment in the molecule which is a sum of the static dipole plus several field dependent term.

:<math>\mu = \mu_0 + (\partial \mu_i / \partial E_j)_{E_0}E_j \quad + \quad 1/2 (\partial^2 \mu_i / \partial E_jE_k)_{E_0} E_jE_k \quad+ \quad 1/6(\partial^3\mu_i / \partial E_jE_kE_j)_{E_0} E_jE_kE_j\,\!</math>

Where
:<math>\mu\,\!</math> is the microscopic nonlinear polarizability
:<math>E_i E_k E_j\,\!</math> are the electric field (vectors)
:<math>\mu = \mu_0 \quad+\quad \alpha_{ij}E_j \quad+\quad \beta _{ijk}/ 2 E E \quad+\quad \gamma_{ijkl} / 6 E E E + ...\,\!</math>

Where:

:<math>\alpha\,\!</math> is linear polarizability
:<math>\beta\,\!</math> is the [[first hyperpolarizability]] ( a third rank tensor with 27 permutations although some are degenerate)
:<math>\gamma\,\!</math> is the [[second hyperpolarizability]], responsible for third order non linear optics.

The terms beyond αE are not linear (they have exponential terms) in E and are therefore referred to as the nonlinear polarization and give rise to nonlinear optical effects. Note that Ej, Ek, and Ej are vectors representing the direction of the polarization of the applied field with respect to the molecular coordinate frame. Molecules are asymmetric have different polarizabilities depending the direction of the applied electric field.

Alpha is the second derivative of the dipole moment with respect to field, and is also the first derivative of the polarizability with respect to field. Beta is the first derivative of polarizability with respect to field, and gamma is the first derivative of the first hyperpolarizability with respect to field.

Nonlinear polarization becomes more important with increasing field strength, since it scales with higher powers of the field (quadratic or cubic relationships). Second harmonic generation was not observed until 1961 after the advent of the laser. Under normal conditions,
:<math>\alpha_{ij}E \quad > \quad \beta_{ijk}/2 E·E \quad > \quad \gamma_{ijkl} /6 E·E·E.\,\!</math>

Thus, there were few observations of NLO effects with normal light before the invention of the laser with its associated large electric fields.

With very large electric fields there can be dielectric breakdown of the material.

The observed bulk polarization density is given by an
expression analogous to (7):

:<math>P = P_o + \chi^{(1)} ·E + \chi^{(2)}·· EE + \chi^{(3)}···EEE+ ...\,\!</math> (8)

where the :<math>\chi^{(i)}\,\!</math> susceptibility coefficients are tensors of order i+1 (e.g., :<math>\chi^{(2)}_{ijk}\,\!</math>).
Po is the intrinsic static dipole moment density of the sample.

The linear polarizability is the ability to polarize a molecule, the linear susceptibility is bulk polarization density in a materials which has to do with the polarizability of the molecules and the density of those molecules in the material. More molecules means a higher susceptibility.

=== Taylor Expansion for Bulk Polarization ===
Consider a simple molecule with all the fields being identical.

:<math>P = P_o + \chi^{(1)} ·E + 1/2\chi^{(2)}·· E^2 + 1/6\chi^{(3)}···E^3+ ...\,\!</math>

In a Taylor series expansion the dots refer the fact that these are tensor products. Just as a molecule can only have a non-zero beta if it is noncentrosymmetric, a material can only have a :<math>\chi^{(2)}\,\!</math> if the material is noncentrosymmetric (i.e., a centrosymmetry arrangement of noncentrosymmetric molecules lead to zero :<math>\chi^{(2)}\,\!</math>) .

In a centrosymmetric material a perturbation by an electric field (E) leads to a polarization P. Therefore, application of an electric field (–E) must lead to a polarization –P.

Now consider the second order polarization in a centrosymmetric material.

:<math>P = \chi^{(2)}·· E^2,\,\!</math> (10)

:<math> –P = \chi^{(2)}·· (–E)^2 = \chi^{(2)}·· E^2\,\!</math> (11)

This only occurs when P = 0, therefore :<math>\chi^{(2)}\,\!</math> must be 0.

This means that if we use quantum mechanics to design molecules that will have large hyperpolarizabilities the effort will be wasted if the molecules arrange themselves in a centrosymmetric manner resulting in bulk susceptibility of zero. The design therefore must include both arranging for the desired electronic properties, but also configuring the molecule so that those molecules will not line up in a centrosymmetric manner in the material. A solution of molecules can also exhibit some centrosymmetry.

== Frequency Effects ==

=== Frequency Doubling and Sum-Frequency Generation ===

One nonlinear optical phenomena is that when you shine light at one frequency on a material you get out light with twice the frequency. This process is known as sum or difference frequency mixing. Two beams with frequency ω1 and ω2 when summed results in a frequency of 2 x ω also referred to as second harmonic generation (SHG) or, if ω1 – ω 2 results in a zero frequency electric field this is a simple voltage also known as optical rectification.

The electronic charge displacement (polarization) induced by an oscillating electric field (e.g., light) can be viewed as a classical oscillating dipole that itself emits radiation at the oscillation frequency.

For linear first-order polarization, the radiation has the same frequency as the incident light.

=== Taylor Expansion with Oscillating Electric Fields-SHG ===

The electric field of a plane light wave can be expressed as

:<math>E = E_0 cos(\omega t)\,\!</math>

Using a power series expansion :<math>Ecos^2(\omega t) E\,\!</math> can be substituted for E

:<math>P = (P_0 + \chi^{(1)}E_0 cos(\omega t) + \chi^{(2)} E_0^2cos^2(\omega t) + \chi^{(3)} E_0^3 cos^3(\omega t) + ...\,\!</math>

Where
:<math>P_0\,\!</math> is the static polarizablity

Since
:<math>cos^2(\omega t)\,\!</math> equals :<math>1/2 + 1/2 cos(2 \omega t)\,\!</math>,

the first three terms of equation (13) become:

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (14)

This is the origin of the process of optical rectification and second harmonic generation.

=== Second Harmonic Generation (SHG) ===

:<math>P = (P^0 + 1/2\chi^{(2)} E_0^2) + \chi^{(1)}E_0cos(\omega t) + 1/2 \chi^{(2)}E_0^2cos(2 \omega t) + ..\,\!</math> (16)

Physically, equation (16) states that the polarization consists of a:

*Second-order DC field contribution to the static polarization (first term),

*Frequency component ω corresponding to the light at the incident frequency (second term) and

*A new frequency doubled component, :<math>2\omega\,\!</math> (third term)-- recall the asymmetric polarization wave and its Fourier analysis.

=== Sum and Difference Frequency Generation ===

In the more general case (in which the two fields are not constrained to be equal), NLO effects involves the interaction of NLO material with two distinct waves with electric fields E1 with the electrons of the NLO material.

Consider two laser beams E1 and E2, the second-order term of equation (4) becomes:
:<math>\chi^{(2)}·E_1cos(\omega_1t)E_2cos(\omega_2t)\,\!</math> (15)

From trigonometry we know that equation (15) is equivalent to:
:<math>1/2\chi^{(2)}·E_1E_2cos [(\omega_1 + \omega_2)t] +1/2\chi^{(2)}·E_1E_2cos [(\omega_1 - \omega_2)t]\,\!</math> (16)

Thus when two light beams of frequencies ω1 and ω2 interact in an NLO material, polarization occurs at sum :<math>(\omega_1 + \omega_2)\,\!</math> and difference :<math>(\omega_1 - \omega_2)\,\!</math> frequencies.

This electronic polarization will therefore, re-emit radiation at these frequencies.

The combination of frequencies is called sum (or difference) frequency generation (SFG) of which SHG is a special case. This is how a tunable laser works.

Note that a very short laser pulse will result in a band or distribution of frequencies due to the Heisenberg Uncertainty Principle. Those bands will add and subtract resulting in some light which is twice the frequency if they added, and some light that is very low frequency (0+ or – the difference), resulting from the difference between the frequencies. This is the process enabling Terahertz spectroscopy. Terahertz is very low frequency light.

Low frequency light is scattered less than high frequency light. For example if you look through a glass of milk there is “index inhomogeneity” in the milk due the presence of protein and fat. Terahertz radiation can be used for surveillance. A terahertz detector scanner will reveal materials that have different index of refraction.

== Electro-optic effects ==

=== Kerr and Pockels Effects ===

John Kerr and Friedrich Pockels discovered in 1875 and 1893, respectively, that the refractive index of a material could be changed by applying a DC or low frequency electric field. This are in fact non-linear optical effects but they often not thought of as such because they don’t require a laser.

Electric impermeability of a material can be expressed as:

:<math>n \equiv \frac {\epsilon_0}{\epsilon} = \frac{1}{n^2}\,\!</math>

:<math>\eta (E) = \eta + rE +SE^2\,\!</math>

Where
:<math>\epsilon_0\,\!</math> is the dielectric constant of free space
:<math>\epsilon\,\!</math> is the dielectric constant

'''Pockels effect'''
[[Image:Pockels_graph.png|thumb|200px|The Pockels effect has a linear relation to applied field]]
In the Pockels effect an applied electric field changes the refractive index of certain materials.

:<math>\eta(E) = \eta - \frac {1} {2}rn^3 E\,\!</math>

Where:

'''r''' is Pockels coefficient or Linear Electro-optic Coefficient, r~ 10-12 – 1—10 m/V, typically.

This is a linear function with respect to the electric field, the higher the r the greater the change. It is cubic with respect to the refractive index so materials with high intrinsic refractive indexes will change more. Some examples include NH4H2PO4(ADP), KH2PO4(KDP), LiNbO3, LiTaO3, CdTe
 
'''The Kerr effect'''
[[Image:Kerr_graph.png|thumb|200px|The Kerr effect has a parabolic relationship to applied field]]
:<math>\eta(E) = \eta – ½ Sn^3E^2\,\!</math>

Where

'''S''' is the Kerr coefficient
*S~ 10-18 – 10-14 mV in crystals
*S~ 10-22 – 10-19 mV in liquids

This is similar to the Pockels effect except that the refractive index varies parabolically or quadratically with the electric field.

This a process that occurs in second order nonlinear optical materials. It is a third order nonlinear optical process. Not all materials are second order nonlinear optical materials, only those that are centrosymmetric. However all materials have a χ(3) even if they are centrosymmetric.

It is possible to change the amplitude, phase or path of light at a given frequency by using a static DC electric field to polarize the material and modify the refractive indices. When light enters a material with a higher refractive index it is phase shifted and the waves become compressed. The direction is also changed. So by changing the refractive index it is possible to change the path of the light.

Consider the special case :<math>\omega_2 = 0\,\!</math> [equation (15)] in which a DC electric field is applied to the material.

The optical frequency polarization (Popt) arising from the second-order susceptibility is given by:

:<math>P_{opt} =\chi^{(2)}·E_1E_2(cos \omega_1 t)\,\!</math> (17)

where:
E1
E2 is the magnitude of the electric field caused by voltage applied to the nonlinear material (a voltage not optical frequency).

Recall that the refractive index is related to the linear susceptibility that is given by the second term of Equation (14):

:<math>\chi^{(1)}·E_1(cos \omega_1 t)\,\!</math> (18)

The total optical frequency polarization (Popt) is the χ(1) term plus the χ(2) term:

:<math>P_{opt} = \chi^{(1)}·E_1(cos_1t) +\chi^{(2)}·E_1E_2(cos \omega_1t)\,\!</math> (19)

Then factor out <math>E_1(cos \omega_1t)\,\!</math> :

:<math> P_{opt} = [\chi^{(1)} + \chi^{(2)}·E_2] E_1(cos \omega_1t)\,\!</math> (20)

χ(1) is linear susceptability which relates to the dielectric constant, which in turn relates to the square of the refractive index. A change in the linear susceptablity changes the index of refraction. The second term: χ(2) times the magnitude of the voltage (E2) means that the susceptability of the material, the dielectric constant of the material, and the refractive index of the material can be altered by changing the applied voltage.

You can shine light on second order nonlinear optical materials and get out different frequencies, or shine one laser beam, apply an electric field and then modulate the refractive index. For example, light can travel freely between two fibers that are very close to each other with the same refractive index. But if the fibers have a different refractive index light will stay in one fiber or the other.

By changing the refractive index you can move light from one fiber to another; it provides a means of switching light in waveguides.

'''Summary'''

*The applied field in effect changes the linear susceptibility and thus the refractive index of the material.

*This is, known as the linear electro-optic (LEO) or Pockels effect, and is used to modulate light by changing the applied voltage.

*At the atomic level, the applied voltage is anisotropically distorting the electron density within the material. Thus, application of a voltage to the material causes the optical beam to "see" a different material with a different polarizability and a different anisotropy of the polarizability than in the absence of the voltage.

*Since the anisotropy is changed upon application of an electric field, a beam of light can have its polarization state (i.e., ellipticity) changed by an amount related to the strength and orientation of the applied voltage, and travel at a different speed and possibly in a different direction.

=== Index modulation ===

Quantitatively, the change in the refractive index as a function of the applied electric field is approximated by
the general expression:

:<math>1/\underline{n}_{ij}2 = 1/n_{ij}2 + r_{ijk}E_k + s_{ijkl}E_kE_l + ... \,\!</math> (21)

where
:<math>\underline{n}_{ij}\,\!</math> are the induced refractive indices,
:<math>n_{ij}\,\!</math> is the refractive index in the absence of the electric field,

:<math>r_{ijk}\,\!</math> is the linear or Pockels coefficients, Δn for E = 106 V/m is 10-6 to 10-4 (crystals) and;

:<math>s_{ijkl}\,\!</math> are the quadratic or Kerr coefficients.

=== r coefficients ===

The optical indicatrix (that characterizes the anisotropy of the refractive index) therefore changes as the electric field within the sample changes. The of map the index of refraction with respect to each polarization of light produces a surface that looks something like a football. The electric field allows you to change the shape of the football.

Electro-optic coefficients are frequently defined in terms of rijk.
The "r" coefficients form a tensor (just as do the coefficient of α).

The subscripts ijk are the same as those used with β. The first subscript (i) refers to the resultant polarization of the material along a defined axis and the following subscripts j and k refer to the orientations of the applied fields, one is the optical frequency field and k is the voltage.

=== Applications of Electro-optic Devices ===
[[Image:Network.png|thumb|400px|EO materials can be used at many locations in a network]]
A network has a variety of devices that provide input from to a transmitter, connected by a electro-optic modulator (EOM) through a switching network, to a receiver with a photodetector, and then are connected to display devices. Nonlinear optical materials can be used for any of these applications. They can used to create terahertz radiation and to create specific wavelengths of light for spectroscopy.
 

[[category:non linear optics]]
[[category:second order NLO]]
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Perturbation Theory

2011-08-05T21:19:57Z

Chrisb: /* Calculation of β components */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. The expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression :<math>e \overrightarrow{r}\,\!</math>. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF - ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole, as depicted in the diagram on the right. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, the system becomes more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

'''First order''': The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

'''Second order''': The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

'''Third order''': For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} + \chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the [[Mathematical Expansion of the Dipole Moment|stark energy expression]] written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the [[Mathematical Expansion of the Dipole Moment|Hellman-Feynman theory]].

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F}) = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 

=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bonds (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alpha_zz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. Beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For γ the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ(3). There is a frequency dependence in χ(3) caused by resonances. This causes spikes in the χ(3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 

=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Perturbation Theory

2011-08-05T19:43:23Z

Chrisb: /* Calculation of β components */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. The expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression :<math>e \overrightarrow{r}\,\!</math>. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF - ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole, as depicted in the diagram on the right. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, the system becomes more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

'''First order''': The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

'''Second order''': The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

'''Third order''': For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} + \chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the [[Mathematical Expansion of the Dipole Moment|stark energy expression]] written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the [[Mathematical Expansion of the Dipole Moment|Hellman-Feynman theory]].

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F}) = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 

=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bonds (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alpha_zz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. Beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 

=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Perturbation Theory

2011-08-05T19:04:49Z

Chrisb: /* Calculation of alpha components */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. The expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression :<math>e \overrightarrow{r}\,\!</math>. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF - ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole, as depicted in the diagram on the right. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, the system becomes more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

'''First order''': The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

'''Second order''': The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

'''Third order''': For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} + \chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the [[Mathematical Expansion of the Dipole Moment|stark energy expression]] written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the [[Mathematical Expansion of the Dipole Moment|Hellman-Feynman theory]].

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F}) = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 

=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bonds (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alpha_zz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Perturbation Theory

2011-08-04T23:35:30Z

Chrisb: /* Perturbation techniques */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. The expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression :<math>e \overrightarrow{r}\,\!</math>. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF - ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole, as depicted in the diagram on the right. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, the system becomes more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

'''First order''': The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

'''Second order''': The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

'''Third order''': For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} + \chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the [[Mathematical Expansion of the Dipole Moment|stark energy expression]] written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the [[Mathematical Expansion of the Dipole Moment|Hellman-Feynman theory]].

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F}) = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 

=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Perturbation Theory

2011-08-03T22:27:02Z

Chrisb: /* Derivative Techniques */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. The expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression :<math>e \overrightarrow{r}\,\!</math>. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF - ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole, as depicted in the diagram on the right. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, the system becomes more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

'''First order''': The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

'''Second order''': The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

'''Third order''': For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} + \chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the [[Mathematical Expansion of the Dipole Moment|stark energy expression]] written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the [[Mathematical Expansion of the Dipole Moment|Hellman-Feynman theory]].

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F}) = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Perturbation Theory

2011-08-03T16:30:38Z

Chrisb: /* Calculation of polarizabilities */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. The expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression :<math>e \overrightarrow{r}\,\!</math>. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF - ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole, as depicted in the diagram on the right. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, the system becomes more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

'''First order''': The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

'''Second order''': The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

'''Third order''': For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} + \chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the [[Mathematical Expansion of the Dipole Moment|stark energy expression]] written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the [[Mathematical Expansion of the Dipole Moment|Hellman-Feynman theory]].

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Perturbation Theory

2011-08-03T16:28:35Z

Chrisb: /* Calculation of polarizabilities */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. The expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression :<math>e \overrightarrow{r}\,\!</math>. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF - ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole, as depicted in the diagram on the right. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, the system becomes more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

'''First order''': The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

'''Second order''': The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

'''Third order''': For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} + \chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the [[Mathematical Expansion of the Dipole Moment|stark energy expression]] written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the Hellman-Feynman theory.

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Perturbation Theory

2011-08-03T16:27:56Z

Chrisb: /* Calculation of polarizabilities */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. The expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression :<math>e \overrightarrow{r}\,\!</math>. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF - ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole, as depicted in the diagram on the right. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, the system becomes more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

'''First order''': The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

'''Second order''': The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

'''Third order''': For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} + \chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the [[stark energy expression|Mathematical Expansion of the Dipole Moment]] written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the Hellman-Feynman theory.

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Perturbation Theory

2011-08-03T16:23:28Z

Chrisb: /* Calculation of polarizabilities */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. The expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression :<math>e \overrightarrow{r}\,\!</math>. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF - ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole, as depicted in the diagram on the right. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, the system becomes more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

'''First order''': The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

'''Second order''': The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

'''Third order''': For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} + \chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the [[Media:stark energy expression]] written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the Hellman-Feynman theory.

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Perturbation Theory

2011-08-03T00:02:08Z

Chrisb: /* Calculation of polarizabilities */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. The expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression :<math>e \overrightarrow{r}\,\!</math>. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF - ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole, as depicted in the diagram on the right. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, the system becomes more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

'''First order''': The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

'''Second order''': The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

'''Third order''': For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} + \chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the stark energy expression written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the Hellman-Feynman theory.

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Perturbation Theory

2011-08-02T23:51:11Z

Chrisb: /* Dipole changes */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. The expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression :<math>e \overrightarrow{r}\,\!</math>. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF - ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole, as depicted in the diagram on the right. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, the system becomes more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

'''First order''': The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

'''Second order''': The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

'''Third order''': For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} +\ chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the stark energy expression written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the Hellman-Feynman theory.

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

<table id="toc" style="width: 100%">
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<td style="text-align: left; width: 33%">[[Mathematical Expansion of the Dipole Moment| Previous Topic]]</td>
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Perturbation Theory

2011-08-02T23:49:18Z

Chrisb: /* Stabilization */

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<td style="text-align: left; width: 33%">[[Mathematical Expansion of the Dipole Moment| Previous Topic]]</td>
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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. The expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression :<math>e \overrightarrow{r}\,\!</math>. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF - ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole, as depicted in the diagram on the right. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, the system becomes more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

First order: The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

Second order:

The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

Third order
For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} +\ chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the stark energy expression written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the Hellman-Feynman theory.

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Perturbation Theory

2011-08-02T23:39:59Z

Chrisb: /* Perturbation theory */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. The expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression :<math>e \overrightarrow{r}\,\!</math>. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF - ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, it gets more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

First order: The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

Second order:

The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

Third order
For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} +\ chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the stark energy expression written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the Hellman-Feynman theory.

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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<td style="text-align: left; width: 33%">[[Mathematical Expansion of the Dipole Moment| Previous Topic]]</td>
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Perturbation Theory

2011-08-02T23:37:15Z

Chrisb: /* Perturbation theory */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. The expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression :<math>e \overrightarrow{r}\,\!</math>. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF - ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves . But it turned out that their guess was wrong. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

It can also be written, as going from right to left. Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, it gets more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

First order: The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

Second order:

The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

Third order
For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} +\ chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the stark energy expression written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the Hellman-Feynman theory.

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Transition Dipole Moment

2011-07-18T22:52:13Z

Chrisb: /* Pseudo-physical description of transition dipole moment */

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Absorption and emission of light require molecules in a material to attain an excited state and then lose energy from the excited state in a specific manner. The transition dipole moment and quantum mechanical rules help predict whether the transition to the excited state is possible or likely.

== Spectroscopy ==
[[Image:Transmittance_spectra.png|thumb|300px|A plot of (I/I0) x 100 vs. Wavelength gives the transmittance spectra]]
When measuring an absorption spectrum a sample is placed in a spectrometer that has a light source with a variable wavelength. The intensity of the light can be measured with a detector as a function of wavelength or as a function of frequency. Typically in spectrometers there are literally two paths: one for reference beam and one for sample beam. The spectrum that is referred to as a transmittance spectrum can be obtained by finding the ratio of the light that is going through the sample, dividing by the light that has been unperturbed, and multiplying that by a hundred. If the molecule does not absorb any light, then all the light goes through and there is a 100% transmittance. If the molecule absorbs 99% of the light, then there is a 1% transmittance. This is one way of doing that and that is referred to as transmittance spectrum.

For reasons of unknown origins, people look at different spectra in different modes. Typically when looking at color, chemists, not physicists, look at spectra in a mode that is referred to as absorbance. But when looking at infrared spectra we typically consider them in terms of transmittance. Absorbance and transmittance are related to one another by the Beer- Lambert law.

:<math>log \frac {l_0}{l} = \varepsilon cl = A\,\!</math>

Where
:<math>A\,\!</math> is the absorbance (a unitless quantity),

:<math>l\,\!</math> is the path length of the sample in cm,

:<math>c\,\!</math> is the concentration of the chromophore in the medium in mol-l-1

:<math>\varepsilon\,\!</math> is the molar absorptivity or extinction coefficient with units of l-mol-1cm-1.

=== Extinction Coefficient ===

The extinction coefficient characterizes the ability of a molecule to absorb light at a given wavelength

The log base 10 of the transmittance is equal to εcl or the absorbance. So the absorbance is simply the log of I0 over I. Keep in mind that absorbance is a unitless quantity. It turns out that absorbance can be related back to some characteristic features of the molecule and that is one of the reasons why it is used today. This formula shows that if there is a certain amount of a molecule and the path length is doubled, twice as much light will be absorbed. At least with the first order, if the concentration of a molecule is doubled, assuming that there are no intermolecular effects, twice as much light will be absorbed. Those are not molecular characteristics; they are characteristics of a sample. ε is related to how well the molecule absorbs light and it is referred to as the molar absorbtivity or the extinction coefficient. Since A (absorbance) is unitless, c is concentration given in moles per liter, and path length is typically given in cm, the units of the extinction coefficient are in: Liter mol-1 cm-1 liter or inverse molar, inverse centimeters. The extinction coefficient characterizes the ability of the molecule to absorb light of the given wavelength. It doesn’t matter whether the sample is liquid or gas. Chemist use log base 10 for the liquid phase, in gas phase spectroscopy you use an absorption cross section with a natural log. Often electrical engineers use natural logs as well.

Try matching elements with the [http://concave.stc.arizona.edu/thepoint/Interactive/spectrasample.html Emission spectra simulator]

=== Oscillator Strength and Molecular Parameters ===

One important question that people ask is “What is the probability my molecule will absorb light?” One way to calculate this is by plotting the absorption spectrum in energy units rather than in wavelength units. The total absorption under the band is expressed with and integral. Note that a scale that is linear in wavelength is not linear in energy.

:<math>f = 4.32 \times 10^9 \int_{-\infty}^{\infty} \varepsilon \partial v\,\!</math>

This integration (dν) gives the '''oscillator strength''', which refers to the probability that a molecule interacting with light over a certain energy range is going to absorb that light.

:<math>f \approx 4.32 \times 10^9 \varepsilon_{max} \Delta v_{1/2}\,\!</math>

That oscillator strength can be approximated by imagining the spectrum as a triangle, and multiplying the extinction at the peak by the width of the band at the half height. This term f can be related back to the transition dipole moment which in turn can be related back to the wave function of the molecules.

==Transition dipole moment ==

If we can say how strong is an absorption band, we can integrate it and extract out the transition dipole moment and that goes right back to the wave function of the molecules, which can be calculated quantum- mechanically. This will allow a chance to think about certain points from what we know about molecular orbital theory.

:<math>f = 4.703 \times 10^{29} \bar{v} \mu^{2}_{ge}\,\!</math>

where

:<math>\bar{v}\,\!</math> is the mean absorption frequency of the band in cm–1

:<math>\mu^{2}_{ge}\,\!</math>, refers to the square of the transition dipole moment between the ground state and the excited state.

=== Dipole moment ===
In classical electrostatics, the energy due to the interaction of an electric field E and a dipole is given by the dot product of the dipole moment and the electric field vector.

:<math>energy = \vec{E} \cdot \vec{\mu}\,\!</math>

The dipole moment is related to the charge on an electron times the difference in charge between two things (so two separated charges that would be two, times their distance.)

:<math>\mu = e_n\Delta q_n r_n\,\!</math>

where:

:<math>e_n\,\!</math> is the elementary charge on the particle n (+ or -).

:<math>\Delta q_n\,\!</math> is the fractional charge.

:<math>r_n\,\!</math> is the distance of that particle from a reference coordinate

In order to get the total dipole moment in a system of many charges you have to sum over all those different vectors.

:<math>\mu_{tot}= \sum_{n} e_n \Delta q_n r_n\,\!</math>

This requires the distance between the charges and the spatial distribution of the charge. Quantum mechanically the probability is related to the square of the wave function. Therefore, to figure out dipole moment using quantum mechanics we keep track of those charges with the wave function. However, we will still have this r term, the distance between them; that is going to tell us what the moment is.

=== Quantum Description of Dipole Moment ===

Consider two electrons in p-orbitals. The distance is 2r between them. The main question here is “Where are the electrons and how can we keep track of them?” In quantum mechanics, the molecular orbital is a linear combination of the atomic orbitals. Therefore, the orbital has a certain description that is given by a linear combination. Take the integral of the wave function of the molecule times the sum of all the different charges, and multiplied again by the complex conjugate of the wave function. These track the position of the electrons in the molecule.

:<math>\mu = \int \Psi^*(\sum_n e_n r_n) \Psi \partial \tau\,\!</math>

Where
:<math>\Psi^*\,\!</math> is the complex conjugate of the wave function

:<math>r_n\,\!</math> is the position of the particle with respect to the coordinate system

:<math>e_n\,\!</math> is the charge on the nth particle

The previous formula can be given more succintly as

:<math>\mu =\int \Psi^* (R) \Psi \partial \tau\,\!</math>

Where

:<math>R\,\!</math> is the charge on the electron times that distance

=== Quantum Description of Transition Dipole Moment ===
It is also possible to obtain a transition dipole moment describing the transition moment between two different states.
The transition dipole moment between two states is the same integral as the previous dipole moment integral except now there is one in the ground state (g) and another in the excited state (e).

:<math>
\mu^{2}_{g,e} \propto \vert \int \Psi^{*}_{g} (R) \Psi_e \partial \tau \vert ^2\,\!</math>

The transition dipole moment involves different states. If they are the same states, for example, one is the ground state and the other is also in the ground state and you have an integral with ψ ground state R times complex conjugate ground state, that integral over all space will tell you the dipole moment. If the integral contains ψ excited state complex conjugate R ψ excited, that integral will tell you the excited state dipole moment. If one is in the ground and one is in the excited state, the integral will tell you the transition moment between the two states. This integral relates the wave function of the molecule back to the oscillator strength and the extinction coefficient.
That integral relates to the strength of the absorption band.

=== Odd and Even Functions and Transition Moments ===
[[Image:Ethylene_mo.png|thumb|300px|In ethylene the excited state must have different symmetry than the ground state.]]

It is important to understand the concept of '''even''' and '''odd''' functions. A function is considered to be even when f(x) = f(-x). An example of an even function is a parabola. An odd function is said to be odd when g(x) = - g(-x). An example of this is the cubic function (y= x3, y=x).

There are a few simple rules:
*The integral over all space for an even function is non zero.
*If you integrate an odd function over all space, that integral will come out to be identically 0.
*An even function times an even function gives an even function.
*An odd function times an odd function gives an even function.
*An even function times an odd function gives an odd function.

Consider the symmetry of ethylene with a π and a π* orbital with a mirror plane between them. The π orbital is even. The π* orbital is odd. The transition dipole moment operator goes as R as distance(i.e. y = x or y = R). The operator function is odd.

Can the integral of the ground state wave function times R times the excited state wave function be non-zero? Since the ground state is even, R is odd, the excited state is odd, the formula contains an even function times an odd function, which is odd, and then odd times odd, is even. Therefore it is possible to be nonzero. However, what if the excited state had the same symmetry as the excited state? For example, the molecular orbitals of butadiene reveal that certain transitions are allowed and certain transitions are forbidden just based on the symmetry.

=== Transition Moment Operator ===
[[Image:Ethylene_mo_transitiondipole.png|thumb|400px|The ground- and excited-state wave function must not have the same symmetry if the transition dipole moment is to be non-zero.]]
Thus, it is should clear that, since R is odd, for molecules where the ground and excited state wave functions are even or odd, the ground- and excited-state wave function must not have the same symmetry if the transition dipole moment is to be non-zero.

Consider the evolution of the wave function for a particle in a box upon excitation from the ground state (with no nodes) to the first excited state with one node.
 

=== Pseudo-physical description of transition dipole moment ===

[[Image:Wavefunction_boxed.png|thumb|300px|Wavefunction during transition]]
Initially the function is symmetric with respect to the axis of the one dimensional box.
In the final state it is also symmetrical, however you can envision a snapshot of the system as the light field is interacting with the wave-function wherein a node begins to develop as is shown in the middle and the wave function is evolving from the initial to final state.
Now consider that the electron density during process is the square of the wave function:

 
[[Image:Electrondensity_boxed.png|thumb|300px|Electron density during transition]]
As can be seen in the initial and final states the electron density is symmetrically distributed with respect to the axis of the box. However with the field on, the electron density is not symmetrically distributed and a transitory dipole moment can be present.

To relate back to real molecules think of each of those orbitals as a linear combination of atomic orbitals. One important factor is the symmetry. But there may be one other factor that will be just as important as symmetry. If you treat orbital 1 as a linear combination over n orbitals and orbital 2 as a linear combinations of orbitals as well, there will be a spatial over lap between the orbital in the ground state and the orbital in the excited state. If there is no spatial overlap between the ground state and excited state orbitals there will be no transition dipole moment. However, if the electrons are in the same place spatially, a large transition dipole moment will result.

Now if the wave functions are considered to be linear combinations of atomic orbitals then:

:<math>\mu^2 _{g,e} \propto \vert \int (\sum_i c_i \Phi_i )^{*}_{g} (R) (\sum_i \Phi_i)_e \partial \tau \vert ^2\,\!</math>

== Summary ==

*There needs to be significant spatial overlap throughout the molecule between the ground- and excited-state wave function, if the transition moment that couples the two states is to be large.

*Because of the position operator, R, if there is good overlap at sites that have a large distance from the origin of the coordinate system, these terms will have enhanced contributions to the transition dipole moment.

*Molecules with good spatial overlap between two states that are not of the same parity (symmetry) will have large transition dipole moments and strong (allowed) transitions in the electronic spectrum

*Conversely, if any of the above criteria are not met, then the transitions will be weak.

 
[[category:absorption]]
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Transition Dipole Moment

2011-07-18T22:22:03Z

Chrisb: /* Transition dipole moment */

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Absorption and emission of light require molecules in a material to attain an excited state and then lose energy from the excited state in a specific manner. The transition dipole moment and quantum mechanical rules help predict whether the transition to the excited state is possible or likely.

== Spectroscopy ==
[[Image:Transmittance_spectra.png|thumb|300px|A plot of (I/I0) x 100 vs. Wavelength gives the transmittance spectra]]
When measuring an absorption spectrum a sample is placed in a spectrometer that has a light source with a variable wavelength. The intensity of the light can be measured with a detector as a function of wavelength or as a function of frequency. Typically in spectrometers there are literally two paths: one for reference beam and one for sample beam. The spectrum that is referred to as a transmittance spectrum can be obtained by finding the ratio of the light that is going through the sample, dividing by the light that has been unperturbed, and multiplying that by a hundred. If the molecule does not absorb any light, then all the light goes through and there is a 100% transmittance. If the molecule absorbs 99% of the light, then there is a 1% transmittance. This is one way of doing that and that is referred to as transmittance spectrum.

For reasons of unknown origins, people look at different spectra in different modes. Typically when looking at color, chemists, not physicists, look at spectra in a mode that is referred to as absorbance. But when looking at infrared spectra we typically consider them in terms of transmittance. Absorbance and transmittance are related to one another by the Beer- Lambert law.

:<math>log \frac {l_0}{l} = \varepsilon cl = A\,\!</math>

Where
:<math>A\,\!</math> is the absorbance (a unitless quantity),

:<math>l\,\!</math> is the path length of the sample in cm,

:<math>c\,\!</math> is the concentration of the chromophore in the medium in mol-l-1

:<math>\varepsilon\,\!</math> is the molar absorptivity or extinction coefficient with units of l-mol-1cm-1.

=== Extinction Coefficient ===

The extinction coefficient characterizes the ability of a molecule to absorb light at a given wavelength

The log base 10 of the transmittance is equal to εcl or the absorbance. So the absorbance is simply the log of I0 over I. Keep in mind that absorbance is a unitless quantity. It turns out that absorbance can be related back to some characteristic features of the molecule and that is one of the reasons why it is used today. This formula shows that if there is a certain amount of a molecule and the path length is doubled, twice as much light will be absorbed. At least with the first order, if the concentration of a molecule is doubled, assuming that there are no intermolecular effects, twice as much light will be absorbed. Those are not molecular characteristics; they are characteristics of a sample. ε is related to how well the molecule absorbs light and it is referred to as the molar absorbtivity or the extinction coefficient. Since A (absorbance) is unitless, c is concentration given in moles per liter, and path length is typically given in cm, the units of the extinction coefficient are in: Liter mol-1 cm-1 liter or inverse molar, inverse centimeters. The extinction coefficient characterizes the ability of the molecule to absorb light of the given wavelength. It doesn’t matter whether the sample is liquid or gas. Chemist use log base 10 for the liquid phase, in gas phase spectroscopy you use an absorption cross section with a natural log. Often electrical engineers use natural logs as well.

Try matching elements with the [http://concave.stc.arizona.edu/thepoint/Interactive/spectrasample.html Emission spectra simulator]

=== Oscillator Strength and Molecular Parameters ===

One important question that people ask is “What is the probability my molecule will absorb light?” One way to calculate this is by plotting the absorption spectrum in energy units rather than in wavelength units. The total absorption under the band is expressed with and integral. Note that a scale that is linear in wavelength is not linear in energy.

:<math>f = 4.32 \times 10^9 \int_{-\infty}^{\infty} \varepsilon \partial v\,\!</math>

This integration (dν) gives the '''oscillator strength''', which refers to the probability that a molecule interacting with light over a certain energy range is going to absorb that light.

:<math>f \approx 4.32 \times 10^9 \varepsilon_{max} \Delta v_{1/2}\,\!</math>

That oscillator strength can be approximated by imagining the spectrum as a triangle, and multiplying the extinction at the peak by the width of the band at the half height. This term f can be related back to the transition dipole moment which in turn can be related back to the wave function of the molecules.

==Transition dipole moment ==

If we can say how strong is an absorption band, we can integrate it and extract out the transition dipole moment and that goes right back to the wave function of the molecules, which can be calculated quantum- mechanically. This will allow a chance to think about certain points from what we know about molecular orbital theory.

:<math>f = 4.703 \times 10^{29} \bar{v} \mu^{2}_{ge}\,\!</math>

where

:<math>\bar{v}\,\!</math> is the mean absorption frequency of the band in cm–1

:<math>\mu^{2}_{ge}\,\!</math>, refers to the square of the transition dipole moment between the ground state and the excited state.

=== Dipole moment ===
In classical electrostatics, the energy due to the interaction of an electric field E and a dipole is given by the dot product of the dipole moment and the electric field vector.

:<math>energy = \vec{E} \cdot \vec{\mu}\,\!</math>

The dipole moment is related to the charge on an electron times the difference in charge between two things (so two separated charges that would be two, times their distance.)

:<math>\mu = e_n\Delta q_n r_n\,\!</math>

where:

:<math>e_n\,\!</math> is the elementary charge on the particle n (+ or -).

:<math>\Delta q_n\,\!</math> is the fractional charge.

:<math>r_n\,\!</math> is the distance of that particle from a reference coordinate

In order to get the total dipole moment in a system of many charges you have to sum over all those different vectors.

:<math>\mu_{tot}= \sum_{n} e_n \Delta q_n r_n\,\!</math>

This requires the distance between the charges and the spatial distribution of the charge. Quantum mechanically the probability is related to the square of the wave function. Therefore, to figure out dipole moment using quantum mechanics we keep track of those charges with the wave function. However, we will still have this r term, the distance between them; that is going to tell us what the moment is.

=== Quantum Description of Dipole Moment ===

Consider two electrons in p-orbitals. The distance is 2r between them. The main question here is “Where are the electrons and how can we keep track of them?” In quantum mechanics, the molecular orbital is a linear combination of the atomic orbitals. Therefore, the orbital has a certain description that is given by a linear combination. Take the integral of the wave function of the molecule times the sum of all the different charges, and multiplied again by the complex conjugate of the wave function. These track the position of the electrons in the molecule.

:<math>\mu = \int \Psi^*(\sum_n e_n r_n) \Psi \partial \tau\,\!</math>

Where
:<math>\Psi^*\,\!</math> is the complex conjugate of the wave function

:<math>r_n\,\!</math> is the position of the particle with respect to the coordinate system

:<math>e_n\,\!</math> is the charge on the nth particle

The previous formula can be given more succintly as

:<math>\mu =\int \Psi^* (R) \Psi \partial \tau\,\!</math>

Where

:<math>R\,\!</math> is the charge on the electron times that distance

=== Quantum Description of Transition Dipole Moment ===
It is also possible to obtain a transition dipole moment describing the transition moment between two different states.
The transition dipole moment between two states is the same integral as the previous dipole moment integral except now there is one in the ground state (g) and another in the excited state (e).

:<math>
\mu^{2}_{g,e} \propto \vert \int \Psi^{*}_{g} (R) \Psi_e \partial \tau \vert ^2\,\!</math>

The transition dipole moment involves different states. If they are the same states, for example, one is the ground state and the other is also in the ground state and you have an integral with ψ ground state R times complex conjugate ground state, that integral over all space will tell you the dipole moment. If the integral contains ψ excited state complex conjugate R ψ excited, that integral will tell you the excited state dipole moment. If one is in the ground and one is in the excited state, the integral will tell you the transition moment between the two states. This integral relates the wave function of the molecule back to the oscillator strength and the extinction coefficient.
That integral relates to the strength of the absorption band.

=== Odd and Even Functions and Transition Moments ===
[[Image:Ethylene_mo.png|thumb|300px|In ethylene the excited state must have different symmetry than the ground state.]]

It is important to understand the concept of '''even''' and '''odd''' functions. A function is considered to be even when f(x) = f(-x). An example of an even function is a parabola. An odd function is said to be odd when g(x) = - g(-x). An example of this is the cubic function (y= x3, y=x).

There are a few simple rules:
*The integral over all space for an even function is non zero.
*If you integrate an odd function over all space, that integral will come out to be identically 0.
*An even function times an even function gives an even function.
*An odd function times an odd function gives an even function.
*An even function times an odd function gives an odd function.

Consider the symmetry of ethylene with a π and a π* orbital with a mirror plane between them. The π orbital is even. The π* orbital is odd. The transition dipole moment operator goes as R as distance(i.e. y = x or y = R). The operator function is odd.

Can the integral of the ground state wave function times R times the excited state wave function be non-zero? Since the ground state is even, R is odd, the excited state is odd, the formula contains an even function times an odd function, which is odd, and then odd times odd, is even. Therefore it is possible to be nonzero. However, what if the excited state had the same symmetry as the excited state? For example, the molecular orbitals of butadiene reveal that certain transitions are allowed and certain transitions are forbidden just based on the symmetry.

=== Transition Moment Operator ===
[[Image:Ethylene_mo_transitiondipole.png|thumb|400px|The ground- and excited-state wave function must not have the same symmetry if the transition dipole moment is to be non-zero.]]
Thus, it is should clear that, since R is odd, for molecules where the ground and excited state wave functions are even or odd, the ground- and excited-state wave function must not have the same symmetry if the transition dipole moment is to be non-zero.

Consider the evolution of the wave function for a particle in a box upon excitation from the ground state (with no nodes) to the first excited state with one node.
 

=== Pseudo-physical description of transition dipole moment ===

[[Image:Wavefunction_boxed.png|thumb|300px|Wavefunction during transition]]
Initially the function is symmetric with respect to the axis of the one dimensional box.
In the final state it is also symmetrical, however you can envision a snapshot of the system as the light field is interacting with the wave-function wherein a node begins to develop as is shown in the middle and the wave function is evolving from the initial to final state.
Now consider that the electron density during process is the square of the wave function:

 
[[Image:Electrondensity_boxed.png|thumb|300px|Electron density during transition]]
As can be seen in the initial and final states the electron density is symmetrically distributed with respect to the axis of the box. However with the field on, the electron density is not symmetrically distributed and a transitory dipole moment can be present.

To relate back to real molecules think of each of those orbitals as a linear combination of atomic orbitals. One important factor is the symmetry. But there may be one other factor that will be just as important as symmetry. If you treat orbital 1 as a linear combination over n orbitals and orbital 2 as a linear combinations of orbitals as well, there will be a spatial over lap between the orbital in the ground state and the orbital in the excited state. If there is no spatial overlap between the ground state and excited state orbitals there will be no transition dipole moment. However, if the electrons are in the same place spatially, a large transition dipole moment will result.

Now if the wave functions are considered to be linear combinations of atomic orbitals then:

:<math>\mu^2 _{g,e} \propto \int \vert (\sum_i c_i \Phi_i )^{*}_{g} (R) (\sum_i \Phi_i)_e \partial \tau \vert ^2\,\!</math>

== Summary ==

*There needs to be significant spatial overlap throughout the molecule between the ground- and excited-state wave function, if the transition moment that couples the two states is to be large.

*Because of the position operator, R, if there is good overlap at sites that have a large distance from the origin of the coordinate system, these terms will have enhanced contributions to the transition dipole moment.

*Molecules with good spatial overlap between two states that are not of the same parity (symmetry) will have large transition dipole moments and strong (allowed) transitions in the electronic spectrum

*Conversely, if any of the above criteria are not met, then the transitions will be weak.

 
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Perturbation Theory

2011-07-15T23:20:38Z

Chrisb: /* Perturbation theory */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. The expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression :<math>e \overrightarrow{r}\,\!</math>. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF minus ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves . But it turned out that their guess was wrong. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

It can also be written, as going from right to left. Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, it gets more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

First order: The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

Second order:

The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

Third order
For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} +\ chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the stark energy expression written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the Hellman-Feynman theory.

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Perturbation Theory

2011-07-15T23:18:40Z

Chrisb: /* Perturbation theory */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. The expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression :<math>\ e \overrightarrow{r}\ !</math>. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF minus ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves . But it turned out that their guess was wrong. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

It can also be written, as going from right to left. Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, it gets more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

First order: The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

Second order:

The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

Third order
For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} +\ chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the stark energy expression written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the Hellman-Feynman theory.

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Perturbation Theory

2011-07-15T22:36:03Z

Chrisb: /* Perturbation theory */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up.
P is summation over all excited states

- μF is the electric dipole approximation. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. This expression here? is the expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ o. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression er. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF minus ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves . But it turned out that their guess was wrong. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

It can also be written, as going from right to left. Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, it gets more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

First order: The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

Second order:

The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

Third order
For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} +\ chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the stark energy expression written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the Hellman-Feynman theory.

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Perturbation Theory

2011-07-15T22:26:20Z

Chrisb: /* Perturbation theory */

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Then it is integrated over space and the complex conjugate is taken. At the second order, the wave functions of the excited state are taken into account to describe the modification of the system. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is -μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in the 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up.
P is summation over all excited states

- μF is the electric dipole approximation. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. This expression here? is the expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ &o;. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression er. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF minus ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves . But it turned out that their guess was wrong. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

It can also be written, as going from right to left. Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, it gets more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

First order: The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

Second order:

The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

Third order
For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} +\ chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the stark energy expression written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the Hellman-Feynman theory.

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Perturbation Theory

2011-07-08T21:47:55Z

Chrisb:

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. At first order, the perturbation is simply acting on the ground state wave function. Then it is integrated over space and the complex conjugate is taken. At second order, the wave functions of the excited state are taken into account to describe the modification of the system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states do play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation and will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is minus μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up.
P is summation over all excited states

- μF is the electric dipole approximation. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. This expression here? is the expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ &o;. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression er. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF minus ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves . But it turned out that their guess was wrong. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

It can also be written, as going from right to left. Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, it gets more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

First order: The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

Second order:

The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

Third order
For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} +\ chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the stark energy expression written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the Hellman-Feynman theory.

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Perturbation Theory

2011-07-08T21:47:27Z

Chrisb:

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. At first order, the perturbation is simply acting on the ground state wave function. Then it is integrated over space and the complex conjugate is taken. At second order, the wave functions of the excited state are taken into account to describe the modification of the system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states do play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation and will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is minus μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up.
P is summation over all excited states

- μF is the electric dipole approximation. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. This expression here? is the expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ &o;. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression er. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF minus ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves . But it turned out that their guess was wrong. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

It can also be written, as going from right to left. Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, it gets more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

First order: The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

Second order:

The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

Third order
For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} +\ chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the stark energy expression written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the Hellman-Feynman theory.

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

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Quantum-Mechanical Theory of Molecular Polarizabilities

2011-07-08T21:46:22Z

Chrisb:

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'''Quantum-Mechanical Theory of Molecular Polarizabilities up to Third Order'''

The literature refers to first order, second order and third order polarizabilities. You may also see “first order hyperpolarizability “ which is the same thing as second order polarizability. These conventions may be confusing.

Our goal is to be able to relate the chemical structure and the nature of the pi-conjugated backbone, and the nature of donor and acceptors to the to kind of non-linear response that can be measured or calculated.

The expression for dipole moment is simply the sum of all the point charges over all those point charges of the charge itself times the position of that charge.

=== Dipole moment ===

 [[Image:Dipole_example.png|thumb|300px|]]
The drawing of the charges shows a point charge of +1 on the left side and a point charge of -1 on the right side. The origin of the system of coordinates is in the middle of these charges. The point charge of +1 is -5 angstrom away from the origin and the point charge of -1 is +5 Ångström away on the x-axis. The first step to calculate the dipole moment is to multiply the charge +1 by the position -5 Angstrom for the +1 point charge, which will be -5eA (electron times angstrom). Then do the same for the -1 point charge (-1 * 5) which equals -5eA as well. Finally the sum of these values (-5eA + -5eA) -10eA, which corresponds to about -48 debyes, will be the dipole moment. The literature does not use the SI units of the dipole moment because the SI units use coulomb for the charge and meters for the distance which are macroscopic units. Instead, the dipole moment units use electron units and Angstrom (eA), or Debye.

:<math>Cb \cdot \, m \, or Debye \approx 4.8 q(\vert e \vert) * d( A )\,\!</math>

If there is a donor acceptor compound, a full charge transfer can occur. The drawing shows the two point charges that are 10 angstrom apart with the length of the conjugated bridge between them. Then with one full electron charge transfer across 10 angstrom, a dipole moment of 50 debyes will be obtained (this is quite large). If the donor and acceptor are about 10 angstrom apart and the dipole moment is about 10 debyes, the system has a charge transfer of about 0.2 electrons from the donor to the acceptor assuming it a simple dipole.

This example is a simplified model where the origin of the system of coordinates was placed midway between the two charges. However, what would happen if the origin of the coordinates is placed on the +1 point charge? What will be the dipole moment? To answer this question, the +1 point charge will be multiplied by a distance of 0 (due to the origin of the coordinates) and that will not contribute anything to the dipole moment. The -1 point charge is now at a distance of 10 angstroms away and is multiplied by a charge of -1 which is -10 eA or about -50 debyes (-48 precisely).
Consider the case where the molecule is not neutral but instead a cation is formed with a +2 charge on the left.
If the origin of the system of the coordinates is at the +2 point charge, the calculated dipole moment will be -10 eA. But if the origin of system coordinate is placed on the -1 point charge, the dipole moment will be -20 eA. An important lesson out of these examples is that with a charged system, the dipole moment depends the choice of origin of coordinates. The dipole moment is no longer perfectly defined.

In the previous example in which the point origin of coordinates was placed on the +1 point charge, there was a net charge so the monopole is not 0. Therefore, the dipole and all the other poles (quadrapole, octapol, hexapole etc.) depend on the origin of the system of coordinates. However, if the monopole is 0, then the dipole doesn’t depend on the origin of coordinates. But if the dipole is not 0, which means there is a permanent dipole moment in the molecule, then the dipole and the other poles will depend on the origin of coordinates. If the monopole is 0 and the dipole is 0, due to the central symmetric molecule, then the quadrapole will not depend on the choice of the origin of the system of coordinates. It is only when the molecule doesn’t carry a net charge that the dipole is really defined. Otherwise, if the molecule does carry a net charge, the dipole is not defined and it will depend on the origin of the system of coordinates that is chosen. This also creates further complications as it is important to make sure that the point of references is the same for similar systems in order to make comparisons. But for now, when one talks about dipoles, the dipoles will be from neutral molecules so that there will be no complications. In the case of two photon absorption molecules the quadrupole moments can be important. But since these molecules are usually centro symmetric and the dipole is 0, the quadrapole will not depend on the choice of the origin of the system of coordinates.

In the case of point charges where it is easy to describe the degree of dependencies or on the system of coordinates.

:<math>\overrightarrow{\mu} = \sum_i q_i \overrightarrow{r_i}\,\!</math> is the dipole for point charges

But if we have a molecule, the Heisenberg principle does not allow a precise location of the electron. Therefore, wave functions are needed. To get the dipole moment, it is necessary to look at the electronic densities at a given point in space. The expression consists of a wave function at a given point in space r (in other words, wave function at r) and the dipole operator. The dipole operator is the electronic charge of the particles being discussed multiplied by the position of those charges. Since r doesn’t modify Ψ, the Ψ can be arranged so that a psi squared is obtained, which is an electronic density. This makes sense because the wave function squared is an electronic density, and so the expression is equal to an electron density multiplied by a position. Those two expressions are perfectly consistent with one another.

:<math>\overrightarrow{\mu} = \int_{-\infty}^{\infty} \Psi^* e \overrightarrow{r} \Psi d \overrightarrow{r}\,\!</math> is the dipole for a molecule

where
:<math>\Psi^*\,\!</math> is the wave function for position r

:<math>e \overrightarrow{r}\,\!</math> is the dipole operator relating the charge and position of charges.

:<math>\langle \Psi \vert \overrightarrow{r} \vert \Psi \rangle\,\!</math> determines the dipole moment in the state described by the wave function Ψ

:<math>\overrightarrow{\mu}_g = \langle \Psi_g \vert \overrightarrow{r} \vert \Psi_g \rangle\,\!</math> This is just the Dirac expression for this integral expression.

The dipole moment can be examined at the excited state and this is critical for second order materials. To find the dipole moment in the excited state S1 of organic chromaphore, use the wave function of that particular excited state.

=== Transition Dipole ===
The dipole moment should not be confused with the transition dipole in which describes the transition from a given state to another state. Usually it is the transition from ground state to an excited state. In that case, the wave function psi g will be that of the initial state and the wave function psi e will be that of the final state.

:<math>\overrightarrow{\mu}_{eg} = \langle \Psi_e | e \overrightarrow {r} | \Psi)g \rangle\,\!</math> the transition dipole

:<math>\overrightarrow{\mu}_{eg} = \int \Psi^*_e e \overrightarrow{r} \Psi _g d\overrightarrow{r}\,\!</math>

Since r, the operator, multiplies but not modifies the wave function, the r term can be rearranged in this way.

:<math>\overrightarrow{\mu}_{eg} = \int \Psi^*_e (\overrightarrow{r}) \Psi _g (\overrightarrow{r}) e\overrightarrow{r} d\overrightarrow{r}\,\!</math>

These wave functions will be key to the description of polarizabilities in molecular compounds.
In this form the equation shows that in order to have large transition dipoles, it is necessary to have a reasonably good wave function overlap between the wave function of the initial state and the wave function of the final state. At the same time, it is also necessary to have that overlap for large values of r.

This means that if the two charges separate due to the polar attraction of the electric field of the light, it is clear that as the distance between two charges get larger, the dipole will become larger as well. Thus, the r value shows that dipole moment will be larger when the charges separate over a larger distance.

=== Polarizablity to polarization ===

Polarizability refers to a molecule and polarization refers to a material. When a molecule with a very large polarizability or material with a very large polarization is place in an electric field the electrons can move a large distance and delocalize a lot resulting in a separation of charges. Therefore, the larger the distance, the larger the state dipole, and the larger the transition dipole.

We will examine the behavior of individual molecule and then see what happens at the level of the material when many molecules come together.

'''Molecular Level'''

At the microscopic (molecular) level, a molecule that can have a permanent dipole moment , mu0 (permanent dipole) will have an induced dipole moment when it is exposed to electric field (or the electric field of the light). That induced dipole moment is proportional to the electric field an alpha (the polarizability) and the electric field. Thus, we can extend the analogy by acknowledging that the larger the alpha, the larger the polarizability, and the larger the induced dipole moment.

:<math>\overrightarrow{\mu} (induced) = \alpha \cdot \overrightarrow{E}\,\!</math>

where

:<math>\overrightarrow{\mu}\,\!</math> the dipole; a vector

:<math>\overrightarrow{E}\,\!</math> the electric field; a vector

:<math>\alpha\,\!</math> the polarizability is a tensor of rank 2

The electric field, the dipole moment, and induced dipole moment are vectors that have x, y, z components but alpha, the polarizability, is actually a tensor. It is a tensor because if an electric field is applied along x, not only can a dipole moment be generated along x but an induced dipole moment can also be generated along y or z. Therefore the induced dipole moment along axis i will depend on all the components of the field and these polarizability tensor components. So by measuring the induced dipole along mux, the x component of the field can also be observed and an alpha xx tensor component will be obtained.

:<math>\overrightarrow{\mu}_i = \sum_j \alpha_{ij} \overrightarrow{E}_j\,\!</math>

Those polarizability tensor components are a function of the wavelength.

:<math>\alpha_{ij} = f(\omega)\,\!</math>

Usually in the literature one talks about wavelengths and then shows the angular frequency, omega, not lambda. This may be confusing. The reason why the angular frequency is used is simply because the wavelength of the light refers to a very specific oscillation frequency of the electric field of the light.

'''Material Level'''

When looking at the macroscopic scale of the molecular material, one talks about the induced polarization of the material. Just like the microscopic scale, the induced polarization will be directly proportional to the electric field at the macroscopic scale.

:<math>\overrightarrow{P}(induced) = \Epsilon_0 \cdot \Chi_{el} \cdot \overrightarrow{E}\,\!</math>

where

:<math>\Epsilon_0\,\!</math> is the electric permittivity of a vacuum

:<math>\Chi_{el}\,\!</math> is the electric susceptibility of the material; a tensor

The displaced electric (D) field takes into account the external field, like the field due to the light and the polarization.

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} + \overrightarrow{P}\,\!</math>

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} + \Epsilon_0 \cdot \Chi_{el} \cdot \overrightarrow{E}\,\!</math>

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} (1+ \Chi)\,\!</math>

The displaced electric field can be expressed as the electric permittivity of the vacuum times the external field times (1 + χ). (1+χ) represents the electric susceptibility, which is an important characteristic of the material. It is also known as the dielectric constant. Dielectric constant is a function of frequency and wavelength and so it varies as a function of wavelength. Thus, it is constant to some extent. The dielectric constant is actually defined as the ratio of the electric permittivity of the medium itself over that of the vacuum.

:<math>\Epsilon_d = \frac {\Epsilon} {\Epsilon_0}\,\!</math> is the dialectric constant

Therefore, the dielectric constant has no dimension since it is a ratio of those two values. By substituting the ratio in the expression for the displaced electric field, a compact expression can be obtained that expresses the displaced electric field as simply the external field times the electric permittivity of the material.

:<math>\overrightarrow{D} = \Epsilon \overrightarrow{E}\,\!</math>

The permittivity and dielectric constant are a function of frequency (ω) or wavelenght (λ).

:<math>\Epsilon_{ij} (\omega) = 1 + \chi_{ij} (\omega)\,\!</math>

=== Relation between dielectric constant and refractive index ===

It is useful to keep in mind the simple relationship between the refractive index and the dielectric constant. Recall the definition of the speed of light, which is 1 over the square root of the electric permittivity of vacuum times the magnetic permeability of vacuum. When light enters a material (a medium), its speed becomes v, which is 1 over the square root of the product of the electric permittivity of that material or medium times the magnetic permittivity of that material or medium. In this case, mu here is the magnetic permittivity; not the dipole moment.

:<math>c = \frac { 1} {\sqrt{\Epsilon_0 \cdot \mu_o}}\,\!</math>

where
:<math>\Epsilon_0 \,\!</math>is the permittivity of vacuum

:<math>\mu_0 \,\!</math>is the magnetic constant or permeability of vacuum

The ratio of the speed of light in vacuum over that in the in the medium is the index of refraction.

:<math>\frac c v = n =\sqrt {\frac {\Epsilon \mu} {\Epsilon_0 {\mu_0}}}\,\!</math>

In the case of weakly magnetic materials μ = μ0

:<math>n(\omega)\approx \sqrt{\Epsilon_D (\omega)}\,\!</math>

The ratio of the speed of light in vacuum over that in the in the medium is the index of refraction.

:<math>n^2(\omega) \approx \Epsilon_D(\omega)\,\!</math>

For materials in which there is no magnetic activity, the magnetic permeability of the material is basically the magnetic permeability of the vacuum. Then the ratio of μ to μ0 will be close to one and thus the index of refraction will be nearly equal to the square root of the electric permittivity of the medium over that of the vacuum; this is the dielectric constant of the material. Often the interaction of the magnetic field with the light of the material is much smaller than the interaction of the electric field. Therefore, in many instances you can neglect the magnetic interactions and this expression would hold.

Eventually, this comes to an expression where the index of refraction squared corresponds to the dielectric constant. This is the connection between the index of refraction and the dielectric constant. It is important to remember that both the index of refraction and the dielectric constant depend on frequency. Also, often many people in the optics or photonics community casually quote the index of refraction for that material or the dielectric constant you will find that when the two numbers are compared, this expression appears to not hold true. This inconsistency is not due to an issue related to the magnetic material but it is simply because when people casually talk about the dielectric constant or index refraction, they are referring to the constant at very small frequency or in a static context where frequency tends to zero.

When people casually talk about the index refraction of the material, they usually mean optical frequencies. Optical frequencies are frequencies that correspond to the visible. Again, when one casually talks about index of refraction, one would imply optical frequencies where as in dielectric constants, one would imply very low frequencies.

Also, note that n, mu and Epsilon are also complex quantities because they may involve both dispersion effects and attenuation effects. As soon as there is absorption, then that becomes an imaginary component to the expression. Thus, all of those are complex numbers.

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2011-07-08T21:45:20Z

Chrisb: /* Quantum Mechanical and Perturbation Theory of Polarizability */

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Dipole Moment

2011-07-08T21:44:26Z

Chrisb: moved Dipole Moment to Mathematical Expansion of the Dipole Moment: Clearer explanation of content

#REDIRECT [[Mathematical Expansion of the Dipole Moment]]

Mathematical Expansion of the Dipole Moment

2011-07-08T21:44:26Z

Chrisb: moved Dipole Moment to Mathematical Expansion of the Dipole Moment: Clearer explanation of content

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Quantum-Mechanical Theory of Molecular Polarizabilities| Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Quantum Mechanical and Perturbation Theory of Polarizability |Return to Quantum Mechanical and Perturbation Theory Menu]]</td>
<td style="text-align: right; width: 33%">[[Perturbation Theory| Next Topic]]</td>
</tr>
</table>
=== Hellman-Feynman Theorem ===

The Hellman-Feynman Theorem, which expresses the dipole moment as minus derivative of the energy of the system with respect to the field. This equation expresses the response of a molecule which is 2nd order in terms of the energy of the molecule and first order in terms of the dipole moment of the molecule (1st order or 2nd order with respect to the field).

:<math>\overrightarrow {\mu} = - \frac {\delta E} {\delta \overrightarrow{F}}\,\!</math>

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} + \frac {1} {2!} \beta \overrightarrow{F}\overrightarrow{F} + \frac {1} {3!} \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\,\!</math>

:<math>\frac {\delta \overrightarrow{\mu}}{\delta \overrightarrow{F}} = \alpha + \beta \overrightarrow{F} + \frac {1} {2!} \gamma \overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

=== Stark Energy Expression ===
Alpha is the linear polarizability or the first order polarizability. It describes how the molecule responds in terms of the modification of its ground state energy or of its dipole moment, in the presence of the field at the limit where the field goes to 0. Thus, α can be cast either as the 1st order derivative of the dipole moment with respect to the field when the field tends to 0 or minus the 2nd order derivative of the ground state energy of the molecule with respect to the field when the field goes to 0.

:<math>\alpha = \left( \frac {\delta\overrightarrow{\mu}}{\delta \overrightarrow{F}}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^2 E}{\delta \overrightarrow{F}^2}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

Beta is the 2nd order derivative of the dipole moment with respect to the field when the field goes to 0. Beta will be referred to as the 2nd order polarizability of the molecule. This can also be derived from the stark energy expression which is the 3rd order derivative of the energy with respect to the field when the field tends to 0.

:<math>\beta = \left( \frac {\delta^2 \overrightarrow{\mu}}{\delta \overrightarrow{F}^2}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^3 E}{\delta \overrightarrow{F}^3}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

Lastly, the γ term corresponds to the 3rd order derivative of the dipole moment or minus the 4th order derivative of the energy with respect to the field at the limit where the field goes to 0. γ will be referred to as the 3rd order polarizability of the molecule.

:<math>\gamma = \left( \frac {\delta^3 \overrightarrow{\mu}}{\delta \overrightarrow{F}^3}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^4 E}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>
Stark energy describes the evolution of the energy of a system of particles in the presence of an electric field F. In the Stark energy expression, γ corresponds to a 4th order term. However the in common terminology α is referred to as the linear polarizability, β the 2nd order polarizability, and γ the 3rd order polarizability. The Hellman-Feynman Theorem is the origin of these terms. Here it is presented as a Taylor series expansion, sometimes one uses a power series expansion.

:<math>
E_g = E^\circ_g = \overrightarrow{\mu}^\circ \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}\overrightarrow{F} - \frac {1} {3!} \beta \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} - \frac {1} {4!} \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\,\!</math>

:<math>= E^\circ_g - \overrightarrow{\mu}\overrightarrow{F}\,\!</math>

=== Electric dipole approximation ===
The stark energy expression states that the ground state energy of the molecule in the presence of the field is the ground state energy in the absence of the field minus μF. Thus, μF is considered the interaction between the field and the molecule. The electric dipole approximation is that only the electric field component of light influences dipole moment. The magnetic component is ignored. In most instances in the literature, this approximation is assumed but not stated. The expression for the dipole moment is expressed as :

<math>\mu^\circ + \alpha F + \beta F^2 ... \,\!</math>

and so on, without paying any attention to where that expression comes from.

:<math>- \overrightarrow{\mu}\overrightarrow{F}\,\!</math>

in dimensional analysis:
:<math>\mu \equiv\,\!</math> charge * distance
:<math>F \equiv\,\!</math> volt/distance
:<math>\mu F \equiv\,\!</math> charge * volt :<math>\equiv\,\!</math> energy

After the electric dipole, the next two terms are the electric quadrupole and the magnetic dipole. 99% of the time only the electric dipole is considered. This represents the difference in energy between the perturbed state and the unperturbed state of the molecules. Thus, this must have an energy dimension. μ is the dipole moment which is the charge times the distance, and the electric field is volt over distance. When you multiply these terms, the expression has the dimension charge times volt or ev, and ev is an energy unit. This will be used in perturbation theory.

The stark energy expression states that the ground state energy of the molecule in the presence of the field is the ground state energy in the absence of the field minus μF. Thus, μF is considered the interaction between the field and the molecule. The electric dipole approximation is that only the electric field component of light influences dipole moment. The magnetic component is ignored. In most instances in the literature, this approximation is assumed but not stated. The expression for the dipole moment is expressed as

:<math>\mu^\circ + \alpha F + \beta F^2...\,\!</math>

and so on, without paying any attention to where that expression comes from.

:<math>\overrightarrow{\mu} = e \sum(i) \overrightarrow{\pi}_i 9i\,\!</math>

After the electric dipole, the next two terms are the electric quadrupole and the magnetic dipole. 99% of the time only the electric dipole is considered. This represents the difference in energy between the perturbed state and the unperturbed state of the molecules. Thus, this must have an energy dimension. μ is the dipole moment which is the charge times the distance, and the electric field is volt over distance. When you multiply these terms, the expression has the dimension charge times volt or ev, and ev is an energy unit. This will be used in perturbation theory.

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Quantum-Mechanical Theory of Molecular Polarizabilities| Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Quantum Mechanical and Perturbation Theory of Polarizability |Return to Quantum Mechanical and Perturbation Theory Menu]]</td>
<td style="text-align: right; width: 33%">[[Perturbation Theory| Next Topic]]</td>
</tr>
</table>

Main Page

2011-07-08T21:30:02Z

Chrisb: /* Quantum Mechanical and Perturbation Theory of Polarizability */

<big>'''Center for Materials and Devices for Information Technology Research (CMDITR) Wiki'''</big> 
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===Quantum Mechanical and Perturbation Theory of Polarizability===
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Main Page

2011-07-08T21:28:01Z

Chrisb: /* Quantum Mechanical and Perturbation Theory of Polarizability */

Mathematical Expansion of the Dipole Moment

2011-07-08T20:16:42Z

Chrisb:

Mathematical Expansion of the Dipole Moment

2011-07-08T20:15:39Z

Chrisb:

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Quantum-Mechanical Theory of Molecular Polarizabilities| Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Quantum Mechanical and Perturbation Theory of Polarizability |Return to Quantum Mechanical and Perturbation Theory Menu]]</td>
<td style="text-align: right; width: 33%">[[Perturbation Theory| Next Topic]]</td>
</tr>
</table>
=== Hellman-Feynman Theorem ===

The Hellman-Feynman Theorem, which expresses the dipole moment as minus derivative of the energy of the system with respect to the field. This equation expresses the response of a molecule which is 2nd order in terms of the energy of the molecule and first order in terms of the dipole moment of the molecule (1st order or 2nd order with respect to the field).

:<math>\overrightarrow {\mu} = - \frac {\delta E} {\delta \overrightarrow{F}}\,\!</math>

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} + \frac {1} {2!} \beta \overrightarrow{F}\overrightarrow{F} + \frac {1} {3!} \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\,\!</math>

:<math>\frac {\delta \overrightarrow{\mu}}{\delta \overrightarrow{F}} = \alpha + \beta \overrightarrow{F} + \frac {1} {2!} \gamma \overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

=== Stark Energy Expression ===
Alpha is the linear polarizability or the first order polarizability. It describes how the molecule responds in terms of the modification of its ground state energy or of its dipole moment, in the presence of the field at the limit where the field goes to 0. Thus, α can be cast either as the 1st order derivative of the dipole moment with respect to the field when the field tends to 0 or minus the 2nd order derivative of the ground state energy of the molecule with respect to the field when the field goes to 0.

:<math>\alpha = \left( \frac {\delta\overrightarrow{\mu}}{\delta \overrightarrow{F}}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^2 \overrightarrow{E}}{\delta \overrightarrow{F}^2}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

Beta is the 2nd order derivative of the dipole moment with respect to the field when the field goes to 0. Beta will be referred to as the 2nd order polarizability of the molecule. This can also be derived from the stark energy expression which is the 3rd order derivative of the energy with respect to the field when the field tends to 0.

:<math>\beta = \left( \frac {\delta^2 \overrightarrow{\mu}}{\delta \overrightarrow{F}^2}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^3 \overrightarrow{E}}{\delta \overrightarrow{F}^3}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

Lastly, the γ term corresponds to the 3rd order derivative of the dipole moment or minus the 4th order derivative of the energy with respect to the field at the limit where the field goes to 0. γ will be referred to as the 3rd order polarizability of the molecule.

:<math>\gamma = \left( \frac {\delta^3 \overrightarrow{\mu}}{\delta \overrightarrow{F}^3}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^4 \overrightarrow{E}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>
Stark energy describes the evolution of the energy of a system of particles in the presence of an electric field F. In the Stark energy expression, γ corresponds to a 4th order term. However the in common terminology α is referred to as the linear polarizability, β the 2nd order polarizability, and γ the 3rd order polarizability. The Hellman-Feynman Theorem is the origin of these terms. Here it is presented as a Taylor series expansion, sometimes one uses a power series expansion.

:<math>
E_g = E^\circ_g = \overrightarrow{\mu}^\circ \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}\overrightarrow{F} - \frac {1} {3!} \beta \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} - \frac {1} {4!} \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\,\!</math>

:<math>= E^\circ_g - \overrightarrow{\mu}\overrightarrow{F}\,\!</math>

=== Electric dipole approximation ===
The stark energy expression states that the ground state energy of the molecule in the presence of the field is the ground state energy in the absence of the field minus μF. Thus, μF is considered the interaction between the field and the molecule. The electric dipole approximation is that only the electric field component of light influences dipole moment. The magnetic component is ignored. In most instances in the literature, this approximation is assumed but not stated. The expression for the dipole moment is expressed as :

<math>\mu^\circ + \alpha F + \beta F^2 ... \,\!</math>

and so on, without paying any attention to where that expression comes from.

:<math>- \overrightarrow{\mu}\overrightarrow{F}\,\!</math>

in dimensional analysis:
:<math>\mu \equiv\,\!</math> charge * distance
:<math>F \equiv\,\!</math> volt/distance
:<math>\mu F \equiv\,\!</math> charge * volt :<math>\equiv\,\!</math> energy

After the electric dipole, the next two terms are the electric quadrupole and the magnetic dipole. 99% of the time only the electric dipole is considered. This represents the difference in energy between the perturbed state and the unperturbed state of the molecules. Thus, this must have an energy dimension. μ is the dipole moment which is the charge times the distance, and the electric field is volt over distance. When you multiply these terms, the expression has the dimension charge times volt or ev, and ev is an energy unit. This will be used in perturbation theory.

The stark energy expression states that the ground state energy of the molecule in the presence of the field is the ground state energy in the absence of the field minus μF. Thus, μF is considered the interaction between the field and the molecule. The electric dipole approximation is that only the electric field component of light influences dipole moment. The magnetic component is ignored. In most instances in the literature, this approximation is assumed but not stated. The expression for the dipole moment is expressed as

:<math>\mu^\circ + \alpha F + \beta F^2...\,\!</math>

and so on, without paying any attention to where that expression comes from.

:<math>\overrightarrow{\mu} = e \sum(i) \overrightarrow{\pi}_i 9i\,\!</math>

After the electric dipole, the next two terms are the electric quadrupole and the magnetic dipole. 99% of the time only the electric dipole is considered. This represents the difference in energy between the perturbed state and the unperturbed state of the molecules. Thus, this must have an energy dimension. μ is the dipole moment which is the charge times the distance, and the electric field is volt over distance. When you multiply these terms, the expression has the dimension charge times volt or ev, and ev is an energy unit. This will be used in perturbation theory.

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Quantum-Mechanical Theory of Molecular Polarizabilities| Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Quantum Mechanical and Perturbation Theory of Polarizability |Return to Quantum Mechanical and Perturbation Theory Menu]]</td>
<td style="text-align: right; width: 33%">[[Perturbation Theory| Next Topic]]</td>
</tr>
</table>

Mathematical Expansion of the Dipole Moment

2011-07-08T20:14:47Z

Chrisb:

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Quantum-Mechanical Theory of Molecular Polarizabilities| Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Quantum Mechanical and Perturbation Theory of Polarizability |Return to Quantum Mechanical and Perturbation Theory Menu]]</td>
<td style="text-align: right; width: 33%">[[Perturbation Theory| Next Topic]]</td>
</tr>
</table>
=== Hellman-Feynman Theorem ===

The Hellman-Feynman Theorem, which expresses the dipole moment as minus derivative of the energy of the system with respect to the field. This equation expresses the response of a molecule which is 2nd order in terms of the energy of the molecule and first order in terms of the dipole moment of the molecule (1st order or 2nd order with respect to the field).

:<math>\overrightarrow {\mu} = - \frac {\delta E} {\delta \overrightarrow{F}}\,\!</math>

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} + \frac {1} {2!} \beta \overrightarrow{F}\overrightarrow{F} + \frac {1} {3!} \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\,\!</math>

:<math>\frac {\delta \overrightarrow{\mu}}{\delta \overrightarrow{F}} = \alpha + \beta \overrightarrow{F} + \frac {1} {2!} \gamma \overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

=== Stark Energy Expression ===
Alpha is the linear polarizability or the first order polarizability. It describes how the molecule responds in terms of the modification of its ground state energy or of its dipole moment, in the presence of the field at the limit where the field goes to 0. Thus, α can be cast either as the 1st order derivative of the dipole moment with respect to the field when the field tends to 0 or minus the 2nd order derivative of the ground state energy of the molecule with respect to the field when the field goes to 0.

:<math>\alpha = \left( \frac {\delta\overrightarrow{\mu}}{\delta \overrightarrow{F}}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^2 \overrightarrow{\E}}{\delta \overrightarrow{F}^2}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

Beta is the 2nd order derivative of the dipole moment with respect to the field when the field goes to 0. Beta will be referred to as the 2nd order polarizability of the molecule. This can also be derived from the stark energy expression which is the 3rd order derivative of the energy with respect to the field when the field tends to 0.

:<math>\beta = \left( \frac {\delta^2 \overrightarrow{\mu}}{\delta \overrightarrow{F}^2}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^3 \overrightarrow{\E}}{\delta \overrightarrow{F}^3}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

Lastly, the γ term corresponds to the 3rd order derivative of the dipole moment or minus the 4th order derivative of the energy with respect to the field at the limit where the field goes to 0. γ will be referred to as the 3rd order polarizability of the molecule.

:<math>\gamma = \left( \frac {\delta^3 \overrightarrow{\mu}}{\delta \overrightarrow{F}^3}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^4 \overrightarrow{\E}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>
Stark energy describes the evolution of the energy of a system of particles in the presence of an electric field F. In the Stark energy expression, γ corresponds to a 4th order term. However the in common terminology α is referred to as the linear polarizability, β the 2nd order polarizability, and γ the 3rd order polarizability. The Hellman-Feynman Theorem is the origin of these terms. Here it is presented as a Taylor series expansion, sometimes one uses a power series expansion.

:<math>
E_g = E^\circ_g = \overrightarrow{\mu}^\circ \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}\overrightarrow{F} - \frac {1} {3!} \beta \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} - \frac {1} {4!} \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\,\!</math>

:<math>= E^\circ_g - \overrightarrow{\mu}\overrightarrow{F}\,\!</math>

=== Electric dipole approximation ===
The stark energy expression states that the ground state energy of the molecule in the presence of the field is the ground state energy in the absence of the field minus μF. Thus, μF is considered the interaction between the field and the molecule. The electric dipole approximation is that only the electric field component of light influences dipole moment. The magnetic component is ignored. In most instances in the literature, this approximation is assumed but not stated. The expression for the dipole moment is expressed as :

<math>\mu^\circ + \alpha F + \beta F^2 ... \,\!</math>

and so on, without paying any attention to where that expression comes from.

:<math>- \overrightarrow{\mu}\overrightarrow{F}\,\!</math>

in dimensional analysis:
:<math>\mu \equiv\,\!</math> charge * distance
:<math>F \equiv\,\!</math> volt/distance
:<math>\mu F \equiv\,\!</math> charge * volt :<math>\equiv\,\!</math> energy

After the electric dipole, the next two terms are the electric quadrupole and the magnetic dipole. 99% of the time only the electric dipole is considered. This represents the difference in energy between the perturbed state and the unperturbed state of the molecules. Thus, this must have an energy dimension. μ is the dipole moment which is the charge times the distance, and the electric field is volt over distance. When you multiply these terms, the expression has the dimension charge times volt or ev, and ev is an energy unit. This will be used in perturbation theory.

The stark energy expression states that the ground state energy of the molecule in the presence of the field is the ground state energy in the absence of the field minus μF. Thus, μF is considered the interaction between the field and the molecule. The electric dipole approximation is that only the electric field component of light influences dipole moment. The magnetic component is ignored. In most instances in the literature, this approximation is assumed but not stated. The expression for the dipole moment is expressed as

:<math>\mu^\circ + \alpha F + \beta F^2...\,\!</math>

and so on, without paying any attention to where that expression comes from.

:<math>\overrightarrow{\mu} = e \sum(i) \overrightarrow{\pi}_i 9i\,\!</math>

After the electric dipole, the next two terms are the electric quadrupole and the magnetic dipole. 99% of the time only the electric dipole is considered. This represents the difference in energy between the perturbed state and the unperturbed state of the molecules. Thus, this must have an energy dimension. μ is the dipole moment which is the charge times the distance, and the electric field is volt over distance. When you multiply these terms, the expression has the dimension charge times volt or ev, and ev is an energy unit. This will be used in perturbation theory.

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Mathematical Expansion of the Dipole Moment

2011-07-08T20:12:54Z

Chrisb:

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=== Hellman-Feynman Theorem ===

The Hellman-Feynman Theorem, which expresses the dipole moment as minus derivative of the energy of the system with respect to the field. This equation expresses the response of a molecule which is 2nd order in terms of the energy of the molecule and first order in terms of the dipole moment of the molecule (1st order or 2nd order with respect to the field).

:<math>\overrightarrow {\mu} = - \frac {\delta E} {\delta \overrightarrow{F}}\,\!</math>

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} + \frac {1} {2!} \beta \overrightarrow{F}\overrightarrow{F} + \frac {1} {3!} \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\,\!</math>

:<math>\frac {\delta \overrightarrow{\mu}}{\delta \overrightarrow{F}} = \alpha + \beta \overrightarrow{F} + \frac {1} {2!} \gamma \overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

=== Stark Energy Expression ===
Alpha is the linear polarizability or the first order polarizability. It describes how the molecule responds in terms of the modification of its ground state energy or of its dipole moment, in the presence of the field at the limit where the field goes to 0. Thus, α can be cast either as the 1st order derivative of the dipole moment with respect to the field when the field tends to 0 or minus the 2nd order derivative of the ground state energy of the molecule with respect to the field when the field goes to 0.

:<math>\alpha = \left( \frac {\delta\overrightarrow{\mu}}{\delta \overrightarrow{F}}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^2 \overrightarrow{\mu}}{\delta \overrightarrow{F}^2}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

Beta is the 2nd order derivative of the dipole moment with respect to the field when the field goes to 0. Beta will be referred to as the 2nd order polarizability of the molecule. This can also be derived from the stark energy expression which is the 3rd order derivative of the energy with respect to the field when the field tends to 0.

:<math>\beta = \left( \frac {\delta^2 \overrightarrow{\mu}}{\delta \overrightarrow{F}^2}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^3 \overrightarrow{\mu}}{\delta \overrightarrow{F}^3}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

Lastly, the γ term corresponds to the 3rd order derivative of the dipole moment or minus the 4th order derivative of the energy with respect to the field at the limit where the field goes to 0. γ will be referred to as the 3rd order polarizability of the molecule.

:<math>\gamma = \left( \frac {\delta^3 \overrightarrow{\mu}}{\delta \overrightarrow{F}^3}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>
Stark energy describes the evolution of the energy of a system of particles in the presence of an electric field F. In the Stark energy expression, γ corresponds to a 4th order term. However the in common terminology α is referred to as the linear polarizability, β the 2nd order polarizability, and γ the 3rd order polarizability. The Hellman-Feynman Theorem is the origin of these terms. Here it is presented as a Taylor series expansion, sometimes one uses a power series expansion.

:<math>
E_g = E^\circ_g = \overrightarrow{\mu}^\circ \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}\overrightarrow{F} - \frac {1} {3!} \beta \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} - \frac {1} {4!} \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\,\!</math>

:<math>= E^\circ_g - \overrightarrow{\mu}\overrightarrow{F}\,\!</math>

=== Electric dipole approximation ===
The stark energy expression states that the ground state energy of the molecule in the presence of the field is the ground state energy in the absence of the field minus μF. Thus, μF is considered the interaction between the field and the molecule. The electric dipole approximation is that only the electric field component of light influences dipole moment. The magnetic component is ignored. In most instances in the literature, this approximation is assumed but not stated. The expression for the dipole moment is expressed as :

<math>\mu^\circ + \alpha F + \beta F^2 ... \,\!</math>

and so on, without paying any attention to where that expression comes from.

:<math>- \overrightarrow{\mu}\overrightarrow{F}\,\!</math>

in dimensional analysis:
:<math>\mu \equiv\,\!</math> charge * distance
:<math>F \equiv\,\!</math> volt/distance
:<math>\mu F \equiv\,\!</math> charge * volt :<math>\equiv\,\!</math> energy

After the electric dipole, the next two terms are the electric quadrupole and the magnetic dipole. 99% of the time only the electric dipole is considered. This represents the difference in energy between the perturbed state and the unperturbed state of the molecules. Thus, this must have an energy dimension. μ is the dipole moment which is the charge times the distance, and the electric field is volt over distance. When you multiply these terms, the expression has the dimension charge times volt or ev, and ev is an energy unit. This will be used in perturbation theory.

The stark energy expression states that the ground state energy of the molecule in the presence of the field is the ground state energy in the absence of the field minus μF. Thus, μF is considered the interaction between the field and the molecule. The electric dipole approximation is that only the electric field component of light influences dipole moment. The magnetic component is ignored. In most instances in the literature, this approximation is assumed but not stated. The expression for the dipole moment is expressed as

:<math>\mu^\circ + \alpha F + \beta F^2...\,\!</math>

and so on, without paying any attention to where that expression comes from.

:<math>\overrightarrow{\mu} = e \sum(i) \overrightarrow{\pi}_i 9i\,\!</math>

After the electric dipole, the next two terms are the electric quadrupole and the magnetic dipole. 99% of the time only the electric dipole is considered. This represents the difference in energy between the perturbed state and the unperturbed state of the molecules. Thus, this must have an energy dimension. μ is the dipole moment which is the charge times the distance, and the electric field is volt over distance. When you multiply these terms, the expression has the dimension charge times volt or ev, and ev is an energy unit. This will be used in perturbation theory.

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Quantum-Mechanical Theory of Molecular Polarizabilities

2011-07-08T19:51:40Z

Chrisb:

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'''Quantum-Mechanical Theory of Molecular Polarizabilities up to Third Order'''

The literature refers to first order, second order and third order polarizabilities. You may also see “first order hyperpolarizability “ which is the same thing as second order polarizability. These conventions may be confusing.

Our goal is to be able to relate the chemical structure and the nature of the pi-conjugated backbone, and the nature of donor and acceptors to the to kind of non-linear response that can be measured or calculated.

The expression for dipole moment is simply the sum of all the point charges over all those point charges of the charge itself times the position of that charge.

=== Dipole moment ===

 [[Image:Dipole_example.png|thumb|300px|]]
The drawing of the charges shows a point charge of +1 on the left side and a point charge of -1 on the right side. The origin of the system of coordinates is in the middle of these charges. The point charge of +1 is -5 angstrom away from the origin and the point charge of -1 is +5 Ångström away on the x-axis. The first step to calculate the dipole moment is to multiply the charge +1 by the position -5 Angstrom for the +1 point charge, which will be -5eA (electron times angstrom). Then do the same for the -1 point charge (-1 * 5) which equals -5eA as well. Finally the sum of these values (-5eA + -5eA) -10eA, which corresponds to about -48 debyes, will be the dipole moment. The literature does not use the SI units of the dipole moment because the SI units use coulomb for the charge and meters for the distance which are macroscopic units. Instead, the dipole moment units use electron units and Angstrom (eA), or Debye.

:<math>Cb \cdot \, m \, or Debye \approx 4.8 q(\vert e \vert) * d( A )\,\!</math>

If there is a donor acceptor compound, a full charge transfer can occur. The drawing shows the two point charges that are 10 angstrom apart with the length of the conjugated bridge between them. Then with one full electron charge transfer across 10 angstrom, a dipole moment of 50 debyes will be obtained (this is quite large). If the donor and acceptor are about 10 angstrom apart and the dipole moment is about 10 debyes, the system has a charge transfer of about 0.2 electrons from the donor to the acceptor assuming it a simple dipole.

This example is a simplified model where the origin of the system of coordinates was placed midway between the two charges. However, what would happen if the origin of the coordinates is placed on the +1 point charge? What will be the dipole moment? To answer this question, the +1 point charge will be multiplied by a distance of 0 (due to the origin of the coordinates) and that will not contribute anything to the dipole moment. The -1 point charge is now at a distance of 10 angstroms away and is multiplied by a charge of -1 which is -10 eA or about -50 debyes (-48 precisely).
Consider the case where the molecule is not neutral but instead a cation is formed with a +2 charge on the left.
If the origin of the system of the coordinates is at the +2 point charge, the calculated dipole moment will be -10 eA. But if the origin of system coordinate is placed on the -1 point charge, the dipole moment will be -20 eA. An important lesson out of these examples is that with a charged system, the dipole moment depends the choice of origin of coordinates. The dipole moment is no longer perfectly defined.

In the previous example in which the point origin of coordinates was placed on the +1 point charge, there was a net charge so the monopole is not 0. Therefore, the dipole and all the other poles (quadrapole, octapol, hexapole etc.) depend on the origin of the system of coordinates. However, if the monopole is 0, then the dipole doesn’t depend on the origin of coordinates. But if the dipole is not 0, which means there is a permanent dipole moment in the molecule, then the dipole and the other poles will depend on the origin of coordinates. If the monopole is 0 and the dipole is 0, due to the central symmetric molecule, then the quadrapole will not depend on the choice of the origin of the system of coordinates. It is only when the molecule doesn’t carry a net charge that the dipole is really defined. Otherwise, if the molecule does carry a net charge, the dipole is not defined and it will depend on the origin of the system of coordinates that is chosen. This also creates further complications as it is important to make sure that the point of references is the same for similar systems in order to make comparisons. But for now, when one talks about dipoles, the dipoles will be from neutral molecules so that there will be no complications. In the case of two photon absorption molecules the quadrupole moments can be important. But since these molecules are usually centro symmetric and the dipole is 0, the quadrapole will not depend on the choice of the origin of the system of coordinates.

In the case of point charges where it is easy to describe the degree of dependencies or on the system of coordinates.

:<math>\overrightarrow{\mu} = \sum_i q_i \overrightarrow{r_i}\,\!</math> is the dipole for point charges

But if we have a molecule, the Heisenberg principle does not allow a precise location of the electron. Therefore, wave functions are needed. To get the dipole moment, it is necessary to look at the electronic densities at a given point in space. The expression consists of a wave function at a given point in space r (in other words, wave function at r) and the dipole operator. The dipole operator is the electronic charge of the particles being discussed multiplied by the position of those charges. Since r doesn’t modify Ψ, the Ψ can be arranged so that a psi squared is obtained, which is an electronic density. This makes sense because the wave function squared is an electronic density, and so the expression is equal to an electron density multiplied by a position. Those two expressions are perfectly consistent with one another.

:<math>\overrightarrow{\mu} = \int_{-\infty}^{\infty} \Psi^* e \overrightarrow{r} \Psi d \overrightarrow{r}\,\!</math> is the dipole for a molecule

where
:<math>\Psi^*\,\!</math> is the wave function for position r

:<math>e \overrightarrow{r}\,\!</math> is the dipole operator relating the charge and position of charges.

:<math>\langle \Psi \vert \overrightarrow{r} \vert \Psi \rangle\,\!</math> determines the dipole moment in the state described by the wave function Ψ

:<math>\overrightarrow{\mu}_g = \langle \Psi_g \vert \overrightarrow{r} \vert \Psi_g \rangle\,\!</math> This is just the Dirac expression for this integral expression.

The dipole moment can be examined at the excited state and this is critical for second order materials. To find the dipole moment in the excited state S1 of organic chromaphore, use the wave function of that particular excited state.

=== Transition Dipole ===
The dipole moment should not be confused with the transition dipole in which describes the transition from a given state to another state. Usually it is the transition from ground state to an excited state. In that case, the wave function psi g will be that of the initial state and the wave function psi e will be that of the final state.

:<math>\overrightarrow{\mu}_{eg} = \langle \Psi_e | e \overrightarrow {r} | \Psi)g \rangle\,\!</math> the transition dipole

:<math>\overrightarrow{\mu}_{eg} = \int \Psi^*_e e \overrightarrow{r} \Psi _g d\overrightarrow{r}\,\!</math>

Since r, the operator, multiplies but not modifies the wave function, the r term can be rearranged in this way.

:<math>\overrightarrow{\mu}_{eg} = \int \Psi^*_e (\overrightarrow{r}) \Psi _g (\overrightarrow{r}) e\overrightarrow{r} d\overrightarrow{r}\,\!</math>

These wave functions will be key to the description of polarizabilities in molecular compounds.
In this form the equation shows that in order to have large transition dipoles, it is necessary to have a reasonably good wave function overlap between the wave function of the initial state and the wave function of the final state. At the same time, it is also necessary to have that overlap for large values of r.

This means that if the two charges separate due to the polar attraction of the electric field of the light, it is clear that as the distance between two charges get larger, the dipole will become larger as well. Thus, the r value shows that dipole moment will be larger when the charges separate over a larger distance.

=== Polarizablity to polarization ===

Polarizability refers to a molecule and polarization refers to a material. When a molecule with a very large polarizability or material with a very large polarization is place in an electric field the electrons can move a large distance and delocalize a lot resulting in a separation of charges. Therefore, the larger the distance, the larger the state dipole, and the larger the transition dipole.

We will examine the behavior of individual molecule and then see what happens at the level of the material when many molecules come together.

'''Molecular Level'''

At the microscopic (molecular) level, a molecule that can have a permanent dipole moment , mu0 (permanent dipole) will have an induced dipole moment when it is exposed to electric field (or the electric field of the light). That induced dipole moment is proportional to the electric field an alpha (the polarizability) and the electric field. Thus, we can extend the analogy by acknowledging that the larger the alpha, the larger the polarizability, and the larger the induced dipole moment.

:<math>\overrightarrow{\mu} (induced) = \alpha \cdot \overrightarrow{E}\,\!</math>

where

:<math>\overrightarrow{\mu}\,\!</math> the dipole; a vector

:<math>\overrightarrow{E}\,\!</math> the electric field; a vector

:<math>\alpha\,\!</math> the polarizability is a tensor of rank 2

The electric field, the dipole moment, and induced dipole moment are vectors that have x, y, z components but alpha, the polarizability, is actually a tensor. It is a tensor because if an electric field is applied along x, not only can a dipole moment be generated along x but an induced dipole moment can also be generated along y or z. Therefore the induced dipole moment along axis i will depend on all the components of the field and these polarizability tensor components. So by measuring the induced dipole along mux, the x component of the field can also be observed and an alpha xx tensor component will be obtained.

:<math>\overrightarrow{\mu}_i = \sum_j \alpha_{ij} \overrightarrow{E}_j\,\!</math>

Those polarizability tensor components are a function of the wavelength.

:<math>\alpha_{ij} = f(\omega)\,\!</math>

Usually in the literature one talks about wavelengths and then shows the angular frequency, omega, not lambda. This may be confusing. The reason why the angular frequency is used is simply because the wavelength of the light refers to a very specific oscillation frequency of the electric field of the light.

'''Material Level'''

When looking at the macroscopic scale of the molecular material, one talks about the induced polarization of the material. Just like the microscopic scale, the induced polarization will be directly proportional to the electric field at the macroscopic scale.

:<math>\overrightarrow{P}(induced) = \Epsilon_0 \cdot \Chi_{el} \cdot \overrightarrow{E}\,\!</math>

where

:<math>\Epsilon_0\,\!</math> is the electric permittivity of a vacuum

:<math>\Chi_{el}\,\!</math> is the electric susceptibility of the material; a tensor

The displaced electric (D) field takes into account the external field, like the field due to the light and the polarization.

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} + \overrightarrow{P}\,\!</math>

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} + \Epsilon_0 \cdot \Chi_{el} \cdot \overrightarrow{E}\,\!</math>

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} (1+ \Chi)\,\!</math>

The displaced electric field can be expressed as the electric permittivity of the vacuum times the external field times (1 + χ). (1+χ) represents the electric susceptibility, which is an important characteristic of the material. It is also known as the dielectric constant. Dielectric constant is a function of frequency and wavelength and so it varies as a function of wavelength. Thus, it is constant to some extent. The dielectric constant is actually defined as the ratio of the electric permittivity of the medium itself over that of the vacuum.

:<math>\Epsilon_d = \frac {\Epsilon} {\Epsilon_0}\,\!</math> is the dialectric constant

Therefore, the dielectric constant has no dimension since it is a ratio of those two values. By substituting the ratio in the expression for the displaced electric field, a compact expression can be obtained that expresses the displaced electric field as simply the external field times the electric permittivity of the material.

:<math>\overrightarrow{D} = \Epsilon \overrightarrow{E}\,\!</math>

The permittivity and dielectric constant are a function of frequency (ω) or wavelenght (λ).

:<math>\Epsilon_{ij} (\omega) = 1 + \chi_{ij} (\omega)\,\!</math>

=== Relation between dielectric constant and refractive index ===

It is useful to keep in mind the simple relationship between the refractive index and the dielectric constant. Recall the definition of the speed of light, which is 1 over the square root of the electric permittivity of vacuum times the magnetic permeability of vacuum. When light enters a material (a medium), its speed becomes v, which is 1 over the square root of the product of the electric permittivity of that material or medium times the magnetic permittivity of that material or medium. In this case, mu here is the magnetic permittivity; not the dipole moment.

:<math>c = \frac { 1} {\sqrt{\Epsilon_0 \cdot \mu_o}}\,\!</math>

where
:<math>\Epsilon_0 \,\!</math>is the permittivity of vacuum

:<math>\mu_0 \,\!</math>is the magnetic constant or permeability of vacuum

The ratio of the speed of light in vacuum over that in the in the medium is the index of refraction.

:<math>\frac c v = n =\sqrt {\frac {\Epsilon \mu} {\Epsilon_0 {\mu_0}}}\,\!</math>

In the case of weakly magnetic materials μ = μ0

:<math>n(\omega)\approx \sqrt{\Epsilon_D (\omega)}\,\!</math>

The ratio of the speed of light in vacuum over that in the in the medium is the index of refraction.

:<math>n^2(\omega) \approx \Epsilon_D(\omega)\,\!</math>

For materials in which there is no magnetic activity, the magnetic permeability of the material is basically the magnetic permeability of the vacuum. Then the ratio of μ to μ0 will be close to one and thus the index of refraction will be nearly equal to the square root of the electric permittivity of the medium over that of the vacuum; this is the dielectric constant of the material. Often the interaction of the magnetic field with the light of the material is much smaller than the interaction of the electric field. Therefore, in many instances you can neglect the magnetic interactions and this expression would hold.

Eventually, this comes to an expression where the index of refraction squared corresponds to the dielectric constant. This is the connection between the index of refraction and the dielectric constant. It is important to remember that both the index of refraction and the dielectric constant depend on frequency. Also, often many people in the optics or photonics community casually quote the index of refraction for that material or the dielectric constant you will find that when the two numbers are compared, this expression appears to not hold true. This inconsistency is not due to an issue related to the magnetic material but it is simply because when people casually talk about the dielectric constant or index refraction, they are referring to the constant at very small frequency or in a static context where frequency tends to zero.

When people casually talk about the index refraction of the material, they usually mean optical frequencies. Optical frequencies are frequencies that correspond to the visible. Again, when one casually talks about index of refraction, one would imply optical frequencies where as in dielectric constants, one would imply very low frequencies.

Also, note that n, mu and Epsilon are also complex quantities because they may involve both dispersion effects and attenuation effects. As soon as there is absorption, then that becomes an imaginary component to the expression. Thus, all of those are complex numbers.

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Quantum-Mechanical Theory of Molecular Polarizabilities

2011-07-08T19:46:33Z

Chrisb:

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'''Quantum-Mechanical Theory of Molecular Polarizabilities up to Third Order'''

The literature refers to first order, second order and third order polarizabilities. You may also see “first order hyperpolarizability “ which is the same thing as second order polarizability. These conventions may be confusing.

Our goal is to be able to relate the chemical structure and the nature of the pi-conjugated backbone, and the nature of donor and acceptors to the to kind of non-linear response that can be measured or calculated.

The expression for dipole moment is simply the sum of all the point charges over all those point charges of the charge itself times the position of that charge.

=== Dipole moment ===

 [[Image:Dipole_example.png|thumb|300px|]]
The drawing of the charges shows a point charge of +1 on the left side and a point charge of -1 on the right side. The origin of the system of coordinates is in the middle of these charges. The point charge of +1 is -5 angstrom away from the origin and the point charge of -1 is +5 Ångström away on the x-axis. The first step to calculate the dipole moment is to multiply the charge +1 by the position -5 Angstrom for the +1 point charge, which will be -5eA (electron times angstrom). Then do the same for the -1 point charge (-1 * 5) which equals -5eA as well. Finally the sum of these values (-5eA + -5eA) -10eA, which corresponds to about -48 debyes, will be the dipole moment. The literature does not use the SI units of the dipole moment because the SI units use coulomb for the charge and meters for the distance which are macroscopic units. Instead, the dipole moment units use electron units and Angstrom (eA), or Debye.

:<math>Cb \cdot \, m \, or Debye \approx 4.8 q(\vert e \vert) * d( A )\,\!</math>

If there is a donor acceptor compound, a full charge transfer can occur. The drawing shows the two point charges that are 10 angstrom apart with the length of the conjugated bridge between them. Then with one full electron charge transfer across 10 angstrom, a dipole moment of 50 debyes will be obtained (this is quite large). If the donor and acceptor are about 10 angstrom apart and the dipole moment is about 10 debyes, the system has a charge transfer of about 0.2 electrons from the donor to the acceptor assuming it a simple dipole.

This example is a simplified model where the origin of the system of coordinates was placed midway between the two charges. However, what would happen if the origin of the coordinates is placed on the +1 point charge? What will be the dipole moment? To answer this question, the +1 point charge will be multiplied by a distance of 0 (due to the origin of the coordinates) and that will not contribute anything to the dipole moment. The -1 point charge is now at a distance of 10 angstroms away and is multiplied by a charge of -1 which is -10 eA or about -50 debyes (-48 precisely).
Consider the case where the molecule is not neutral but instead a cation is formed with a +2 charge on the left.
If the origin of the system of the coordinates is at the +2 point charge, the calculated dipole moment will be -10 eA. But if the origin of system coordinate is placed on the -1 point charge, the dipole moment will be -20 eA. An important lesson out of these examples is that with a charged system, the dipole moment depends the choice of origin of coordinates. The dipole moment is no longer perfectly defined.

In the previous example in which the point origin of coordinates was placed on the +1 point charge, there was a net charge so the monopole is not 0. Therefore, the dipole and all the other poles (quadrapole, octapol, hexapole etc.) depend on the origin of the system of coordinates. However, if the monopole is 0, then the dipole doesn’t depend on the origin of coordinates. But if the dipole is not 0, which means there is a permanent dipole moment in the molecule, then the dipole and the other poles will depend on the origin of coordinates. If the monopole is 0 and the dipole is 0, due to the central symmetric molecule, then the quadrapole will not depend on the choice of the origin of the system of coordinates. It is only when the molecule doesn’t carry a net charge that the dipole is really defined. Otherwise, if the molecule does carry a net charge, the dipole is not defined and it will depend on the origin of the system of coordinates that is chosen. This also creates further complications as it is important to make sure that the point of references is the same for similar systems in order to make comparisons. But for now, when one talks about dipoles, the dipoles will be from neutral molecules so that there will be no complications. In the case of two photon absorption molecules the quadrupole moments can be important. But since these molecules are usually centro symmetric and the dipole is 0, the quadrapole will not depend on the choice of the origin of the system of coordinates.

In the case of point charges where it is easy to describe the degree of dependencies or on the system of coordinates.

:<math>\overrightarrow{\mu} = \sum_i q_i \overrightarrow{r_i}\,\!</math> is the dipole for point charges

But if we have a molecule, the Heisenberg principle does not allow a precise location of the electron. Therefore, wave functions are needed. To get the dipole moment, it is necessary to look at the electronic densities at a given point in space. The expression consists of a wave function at a given point in space r (in other words, wave function at r) and the dipole operator. The dipole operator is the electronic charge of the particles being discussed multiplied by the position of those charges. Since r doesn’t modify Ψ, the Ψ can be arranged so that a psi squared is obtained, which is an electronic density. This makes sense because the wave function squared is an electronic density, and so the expression is equal to an electron density multiplied by a position. Those two expressions are perfectly consistent with one another.

:<math>\overrightarrow{\mu} = \int_{-\infty}^{\infty} \Psi^* e \overrightarrow{r} \Psi d \overrightarrow{r}\,\!</math> is the dipole for a molecule

where
:<math>\Psi^*\,\!</math> is the wave function for position r

:<math>e \overrightarrow{r}\,\!</math> is the dipole operator relating the charge and position of charges.

:<math>\langle \Psi \vert \overrightarrow{r} \vert \Psi \rangle\,\!</math> determines the dipole moment in the state described by the wave function Ψ

:<math>\overrightarrow{\mu}_g = \langle \Psi_g \vert \overrightarrow{r} \vert \Psi_g \rangle\,\!</math> This is just the Dirac expression for this integral expression.

The dipole moment can be examined at the excited state and this is critical for second order materials. To find the dipole moment in the excited state S1 of organic chromaphore, use the wave function of that particular excited state.

=== Transition Dipole ===
The dipole moment should not be confused with the transition dipole in which describes the transition from a given state to another state. Usually it is the transition from ground state to an excited state. In that case, the wave function psi g will be that of the initial state and the wave function psi e will be that of the final state.

:<math>\overrightarrow{\mu}_{eg} = \langle \Psi_e | e \overrightarrow {r} | \Psi)g \rangle\,\!</math> the transition dipole

:<math>\overrightarrow{\mu}_{eg} = \int \Psi^*_e e \overrightarrow{r} \Psi _g d\overrightarrow{r}\,\!</math>

Since r, the operator, multiplies but not modifies the wave function, the r term can be rearranged in this way.

:<math>\overrightarrow{\mu}_{eg} = \int \Psi^*_e (\overrightarrow{r}) \Psi _g (\overrightarrow{r}) e\overrightarrow{r} d\overrightarrow{r}\,\!</math>

These wave functions will be key to the description of polarizabilities in molecular compounds.
In this form the equation shows that in order to have large transition dipoles, it is necessary to have a reasonably good wave function overlap between the wave function of the initial state and the wave function of the final state. At the same time, it is also necessary to have that overlap for large values of r.

This means that if the two charges separate due to the polar attraction of the electric field of the light, it is clear that as the distance between two charges get larger, the dipole will become larger as well. Thus, the r value shows that dipole moment will be larger when the charges separate over a larger distance.

=== Polarizablity to polarization ===

Polarizability refers to a molecule and polarization refers to a material. When a molecule with a very large polarizability or material with a very large polarization is place in an electric field the electrons can move a large distance and delocalize a lot resulting in a separation of charges. Therefore, the larger the distance, the larger the state dipole, and the larger the transition dipole.

We will examine the behavior of individual molecule and then see what happens at the level of the material when many molecules come together.

'''Molecular Level'''

At the microscopic (molecular) level, a molecule that can have a permanent dipole moment , mu0 (permanent dipole) will have an induced dipole moment when it is exposed to electric field (or the electric field of the light). That induced dipole moment is proportional to the electric field an alpha (the polarizability) and the electric field. Thus, we can extend the analogy by acknowledging that the larger the alpha, the larger the polarizability, and the larger the induced dipole moment.

:<math>\overrightarrow{\mu} (induced) = \alpha \cdot \overrightarrow{E}\,\!</math>

where

:<math>\overrightarrow{\mu}\,\!</math> the dipole; a vector

:<math>\overrightarrow{E}\,\!</math> the electric field; a vector

:<math>\alpha\,\!</math> the polarizability is a tensor of rank 2

The electric field, the dipole moment, and induced dipole moment are vectors that have x, y, z components but alpha, the polarizability, is actually a tensor. It is a tensor because if an electric field is applied along x, not only can a dipole moment be generated along x but an induced dipole moment can also be generated along y or z. Therefore the induced dipole moment along axis i will depend on all the components of the field and these polarizability tensor components. So by measuring the induced dipole along mux, the x component of the field can also be observed and an alpha xx tensor component will be obtained.

:<math>\overrightarrow{\mu}_i = \sum_j \alpha_{ij} \overrightarrow{E}_j\,\!</math>

Those polarizability tensor components are a function of the wavelength.

:<math>\alpha_{ij} = f(\omega)\,\!</math>

Usually in the literature one talks about wavelengths and then shows the angular frequency, omega, not lambda. This may be confusing. The reason why the angular frequency is used is simply because the wavelength of the light refers to a very specific oscillation frequency of the electric field of the light.

'''Material Level'''

When looking at the macroscopic scale of the molecular material, one talks about the induced polarization of the material. Just like the microscopic scale, the induced polarization will be directly proportional to the electric field at the macroscopic scale.

:<math>\overrightarrow{P}(induced) = \Epsilon_0 \cdot \Chi_{el} \cdot \overrightarrow{E}\,\!</math>

where

:<math>\Epsilon_0\,\!</math> is the electric permittivity of a vacuum

:<math>\Chi_{el}\,\!</math> is the electric susceptibility of the material; a tensor

The displaced electric (D) field takes into account the external field, like the field due to the light and the polarization.

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} + \overrightarrow{P}\,\!</math>

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} + \Epsilon_0 \cdot \Chi_{el} \cdot \overrightarrow{E}\,\!</math>

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} (1+ \Chi)\,\!</math>

The displaced electric field can be expressed as the electric permittivity of the vacuum times the external field times (1 + χ). (1+χ) represents the electric susceptibility, which is an important characteristic of the material. It is also known as the dielectric constant. Dielectric constant is a function of frequency and wavelength and so it varies as a function of wavelength. Thus, it is constant to some extent. The dielectric constant is actually defined as the ratio of the electric permittivity of the medium itself over that of the vacuum.

:<math>\Epsilon_d = \frac {\Epsilon} {\Epsilon_0}\,\!</math> is the dialectric constant

Therefore, the dielectric constant has no dimension since it is a ratio of those two values. By substituting the ratio in the expression for the displaced electric field, a compact expression can be obtained that expresses the displaced electric field as simply the external field times the electric permittivity of the material.

:<math>\overrightarrow{D} = \Epsilon \overrightarrow{E}\,\!</math>

The permittivity and dielectric constant are a function of frequency (ω) or wavelenght (λ).

:<math>\Epsilon_{ij} (\omega) = 1 + \chi_{ij} (\omega)\,\!</math>

=== Relation between dielectric constant and refractive index ===

It is useful to keep in mind the simple relationship between the refractive index and the dielectric constant. Recall the definition of the speed of light, which is 1 over the square root of the electric permittivity of vacuum times the magnetic permeability of vacuum. When light enters a material (a medium), its speed becomes v, which is 1 over the square root of the product of the electric permittivity of that material or medium times the magnetic permittivity of that material or medium. In this case, mu here is the magnetic permittivity; not the dipole moment.

:<math>c = \frac { 1} {\sqrt{\Epsilon_0 \cdot \mu_o}}\,\!</math>

where
:<math>\Epsilon_0 \,\!</math>is the permittivity of vacuum

:<math>\mu_0 \,\!</math>is the magnetic constant or permeability of vacuum

The ratio of the speed of light in vacuum over that in the in the medium is the index of refraction.

:<math>\frac c v = n =\sqrt {\frac {\Epsilon \mu} {\Epsilon_0 {\mu_0}}}\,\!</math>

In the case of weakly magnetic materials μ = μ0

:<math>n(\omega)\approx \sqrt{\Epsilon_D (\omega)}\,\!</math>

The ratio of the speed of light in vacuum over that in the in the medium is the index of refraction.

:<math>n^2(\omega) \approx \Epsilon_D(\omega)\,\!</math>

For materials in which there is no magnetic activity, the magnetic permeability of the material is basically the magnetic permeability of the vacuum. Then the ratio of μ to μ0 will be close to one and thus the index of refraction will be nearly equal to the square root of the electric permittivity of the medium over that of the vacuum; this is the dielectric constant of the material. Often the interaction of the magnetic field with the light of the material is much smaller than the interaction of the electric field. Therefore, in many instances you can neglect the magnetic interactions and this expression would hold.

Eventually, this comes to an expression where the index of refraction squared corresponds to the dielectric constant. This is the connection between the index of refraction and the dielectric constant. It is important to remember that both the index of refraction and the dielectric constant depend on frequency. Also, often many people in the optics or photonics community casually quote the index of refraction for that material or the dielectric constant you will find that when the two numbers are compared, this expression appears to not hold true. This inconsistency is not due to an issue related to the magnetic material but it is simply because when people casually talk about the dielectric constant or index refraction, they are referring to the constant at very small frequency or in a static context where frequency tends to zero.

When people casually talk about the index refraction of the material, they usually mean optical frequencies. Optical frequencies are frequencies that correspond to the visible. Again, when one casually talks about index of refraction, one would imply optical frequencies where as in dielectric constants, one would imply very low frequencies.

Also, note that n, mu and Epsilon are also complex quantities because they may involve both dispersion effects and attenuation effects. As soon as there is absorption, then that becomes an imaginary component to the expression. Thus, all of those are complex numbers.

Perturbation Theory

2011-06-28T21:42:19Z

Chrisb:

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=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. At first order, the perturbation is simply acting on the ground state wave function. Then it is integrated over space and the complex conjugate is taken. At second order, the wave functions of the excited state are taken into account to describe the modification of the system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states do play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation and will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is minus μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up.
P is summation over all excited states

- μF is the electric dipole approximation. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. This expression here? is the expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ &o;. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression er. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF minus ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves . But it turned out that their guess was wrong. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

It can also be written, as going from right to left. Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, it gets more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

First order: The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

Second order:

The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

Third order
For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} +\ chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the stark energy expression written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the Hellman-Feynman theory.

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

Perturbation Theory

2011-06-28T21:27:20Z

Chrisb:

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=== Hellman-Feynman Theorem ===

The Hellman-Feynman Theorem, which expresses the dipole moment as minus derivative of the energy of the system with respect to the field. This equation expresses the response of a molecule which is 2nd order in terms of the energy of the molecule and first order in terms of the dipole moment of the molecule (1st order or 2nd order with respect to the field).

:<math>\overrightarrow {\mu} = - \frac {\delta E} {\delta \overrightarrow{F}}\,\!</math>

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} + \frac {1} {2!} \beta \overrightarrow{F}\overrightarrow{F} + \frac {1} {3!} \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\,\!</math>

:<math>\frac {\delta \overrightarrow{\mu}}{\delta \overrightarrow{F}} = \alpha + \beta \overrightarrow{F} + \frac {1} {2!} \gamma \overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

=== Stark Energy Expression ===
Alpha is the linear polarizability or the first order polarizability. It describes how the molecule responds in terms of the modification of its ground state energy or of its dipole moment, in the presence of the field at the limit where the field goes to 0. Thus, α can be cast either as the 1st order derivative of the dipole moment with respect to the field when the field tends to 0 or minus the 2nd order derivative of the ground state energy of the molecule with respect to the field when the field goes to 0.

:<math>\alpha = \left( \frac {\delta\overrightarrow{\mu}}{\delta \overrightarrow{F}}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^2 \overrightarrow{\mu}}{\delta \overrightarrow{F}^2}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

Beta is the 2nd order derivative of the dipole moment with respect to the field when the field goes to 0. Beta will be referred to as the 2nd order polarizability of the molecule. This can also be derived from the stark energy expression which is the 3rd order derivative of the energy with respect to the field when the field tends to 0.

:<math>\beta = \left( \frac {\delta^2 \overrightarrow{\mu}}{\delta \overrightarrow{F}^2}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^3 \overrightarrow{\mu}}{\delta \overrightarrow{F}^3}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

Lastly, the γ term corresponds to the 3rd order derivative of the dipole moment or minus the 4th order derivative of the energy with respect to the field at the limit where the field goes to 0. γ will be referred to as the 3rd order polarizability of the molecule.

:<math>\gamma = \left( \frac {\delta^3 \overrightarrow{\mu}}{\delta \overrightarrow{F}^3}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>
Stark energy describes the evolution of the energy of a system of particles in the presence of an electric field F. In the Stark energy expression, γ corresponds to a 4th order term. However the in common terminology α is referred to as the linear polarizability, β the 2nd order polarizability, and γ the 3rd order polarizability. The Hellman-Feynman Theorem is the origin of these terms. Here it is presented as a Taylor series expansion, sometimes one uses a power series expansion.

:<math>
E_g = E^\circ_g = \overrightarrow{\mu}^\circ \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}\overrightarrow{F} - \frac {1} {3!} \beta \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} - \frac {1} {4!} \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\,\!</math>

:<math>= E^\circ_g - \overrightarrow{\mu}\overrightarrow{F}\,\!</math>

=== Electric dipole approximation ===
The stark energy expression states that the ground state energy of the molecule in the presence of the field is the ground state energy in the absence of the field minus μF. Thus, μF is considered the interaction between the field and the molecule. The electric dipole approximation is that only the electric field component of light influences dipole moment. The magnetic component is ignored. In most instances in the literature, this approximation is assumed but not stated. The expression for the dipole moment is expressed as :

<math>\mu^\circ + \alpha F + \beta F^2 ... \,\!</math>

and so on, without paying any attention to where that expression comes from.

:<math>- \overrightarrow{\mu}\overrightarrow{F}\,\!</math>

in dimensional analysis:
:<math>\mu \equiv\,\!</math> charge * distance
:<math>F \equiv\,\!</math> volt/distance
:<math>\mu F \equiv\,\!</math> charge * volt :<math>\equiv\,\!</math> energy

After the electric dipole, the next two terms are the electric quadrupole and the magnetic dipole. 99% of the time only the electric dipole is considered. This represents the difference in energy between the perturbed state and the unperturbed state of the molecules. Thus, this must have an energy dimension. μ is the dipole moment which is the charge times the distance, and the electric field is volt over distance. When you multiply these terms, the expression has the dimension charge times volt or ev, and ev is an energy unit. This will be used in perturbation theory.

The stark energy expression states that the ground state energy of the molecule in the presence of the field is the ground state energy in the absence of the field minus μF. Thus, μF is considered the interaction between the field and the molecule. The electric dipole approximation is that only the electric field component of light influences dipole moment. The magnetic component is ignored. In most instances in the literature, this approximation is assumed but not stated. The expression for the dipole moment is expressed as

:<math>\mu^\circ + \alpha F + \beta F^2...\,\!</math>

and so on, without paying any attention to where that expression comes from.

:<math>\overrightarrow{\mu} = e \sum(i) \overrightarrow{\pi}_i 9i\,\!</math>

After the electric dipole, the next two terms are the electric quadrupole and the magnetic dipole. 99% of the time only the electric dipole is considered. This represents the difference in energy between the perturbed state and the unperturbed state of the molecules. Thus, this must have an energy dimension. μ is the dipole moment which is the charge times the distance, and the electric field is volt over distance. When you multiply these terms, the expression has the dimension charge times volt or ev, and ev is an energy unit. This will be used in perturbation theory.

=== Perturbation theory ===
The perturbation theory will not be discussed at the quantum- mechanics level. The energy of the ground state of the system is the energy for the unperturbed system. Perturbation can be observed at different orders. At first order, the perturbation is referred as w. If that perturbation is the impact of the electric field of the light in the electric dipole approximation, the perturbation can be expressed as minus μF.

At the first order, the perturbation is operating on the unperturbed wave function of the ground state. Perturbation theory involves modification of systems due to the perturbation of all the wave functions for the unperturbed system. At first order, the perturbation is simply acting on the ground state wave function. Then it is integrated over space and the complex conjugate is taken. At second order, the wave functions of the excited state are taken into account to describe the modification of the system. An unperturbed system has a well defined wave function for the ground state and well defined wave functions for the excited states. The perturbed system is described on the basis of the wave functions of the unperturbed system.
:<math>=\int \Psi* e \overrightarrow{\pi} \Psi dr\,\!</math>

:<math>E_g = E^\circ_g + \langle \Psi_g | w | \Psi_g \rangle + \sum_p \frac {\langle \Psi_g | W | \Psi_p \rangle \langle \Psi_p | W | \Psi_g \rangle + ...}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>E_g = E^\circ_g -\overrightarrow{\mu ^\circ} \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}^2 - ....\,\!</math>

:<math>W = - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

In terms of non-linear optics, the perturbation theory expressions will show what the excited states are in your isolated molecule that will contribute to the linear polarizability, 2nd order polarizability, or the 3rd order polarizability and allow you to pinpoint exactly what excited states do play a major role in your optical response.

The complete set of wave functions for the unperturbed state will form the basic set for the perturbation expressions. In principle this includes all excited states. The 2nd order term in terms of perturbation and will correspond to α, the linear polarizability. In most conjugated systems, only the first excited state needs to be examined. This will often be the case for α and π-conjugated systems as well as for β. But not for γ in which two or more excited states must be taken into account. However, the number of states that need close attention can be heavily restricted.

:<math>E_g = E_g^\circ - \underbrace{\langle | \Psi_g | W | \Psi _g \rangle} \overrightarrow{F} + \,\!</math>

::::<math>\overrightarrow{\mu^\circ}\,\!</math>

:<math>\underbrace{\sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}}\,\!</math>

:::::<math>-\frac {1} {2!}\alpha\,\!</math>

A one to one correspondence can be made between the terms in the stark energy expression and the perturbation theory expression when the perturbation is minus μF.

As a side note, as you go to higher orders, things will look a bit more complicated because there are more summations over excited states. For example in 3rd order, there will be a double summation over excited states. In 4th order, there will be a triple summation over excited states. But it will always be products of matrix elements of this kind at the numerator, and the differences in energies of the states for the unperturbed molecule will be in the denominator. The expressions look more complex but by looking at the terms individually, notice that the same kind of terms come up.
P is summation over all excited states

- μF is the electric dipole approximation. The operator expression, μ is the unit electric charge times the position, which is the dipole moment. The electric field does not do anything to the wave functions of the unperturbed state. This expression here? is the expression of the dipole moment for the ground state Φg is the wave function of the ground state in the unperturbed state, which is simply μ &o;. At first order, the goal is to find the possible dipole moment. If there is a central symmetry, there won’t be any permanent dipole moment of the molecule. If there is a permanent dipole moment, there will be an interaction between that permanent dipole moment and the external field. At second order, the expression includes the summation over all the excited states p. Here perturbation is replaced by its expression er. Since this deals with the wave functions of the unperturbed system, the electric field is outside. This shows a transition dipole between the ground state and excited state p. and the transition dipole between excited state p and the ground state. These terms are equal. They are the exact same transition dipole. The denominator is the square of the transition dipole between the ground state and excited state p.

The numerator is the difference in energy between the ground state energy and the excited state energy.
Finally, by closely examining the stark energy expression, a connection can be made between the term that is linear in the field, μoF minus ½ α. This shows the expression for α as a function of this perturbation expression.

:<math>\alpha=2 \sum_p \frac {\langle \Psi_g | e \overrightarrow{r} | \Psi_p \rangle \langle \Psi_p | e \overrightarrow{r} | \Psi_g \rangle \overrightarrow{F}\overrightarrow{F}}{ E^\circ_p - E^\circ _g}\,\!</math>

:<math>=2 \sum \frac {M_{gp} ^2} {E^\circ _{gp}}\,\!</math>

In this expression there is no longer a minus sign because the denominator is reversed; E of Pnot – E of gnot.

Now there is a compact expression where α is equal to 2 times the summation over all excited states of transition dipole with state p times transition dipole with state p over the transition energy. Taking into account of the perturbation theory, α, the linear polarizability, can be described as a sum over all excited states of the square of the transition dipole between the ground state and the excited state, over the transition energy from the ground state.

[[Image:Perturblevels.png|thumb|300px|]]

A pictorial description shows the process of going from the ground state to excited state p and that is a transition dipole between g and p. There is also another transition dipole when coming back from p to g. That is why the expression for α shows transition dipole squared. As previously explained, the expressions for β will look more complex due to the double summations over excited states. The expression for γ will look even more complex due to the triple summations over excited states. However for all instances, the numerator will always be products of transition dipoles and the denominator will contain the transition energy. In the literature, the perturbation theory expressions are also referred to as “sum over states expressions” the expression contains the sum over all excited states.

A few important questions include “What is the impact of the perturbation on the energy of the system?” and “Would it stabilize or destabilize the system when looking at the perturbation at different orders?”.
It is crucial to understand the differences and variations in conventions. Suppose you want to calculate the dipole moment of the molecule using two programs. First, you input the geometry of the molecule exactly in the same way for both programs. Then, you run the calculation. One program gave a dipole moment of +1.3 Debye, but the other program gave you a dipole moment of -1.3 Debye. Why is there a difference? The difference occurs because the conventions are different for chemists and physicists.

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>\overrightarrow{\mu}= \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F}+ 1/2 \beta \overrightarrow{F}\overrightarrow{F} +1/6 \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} ...\,\!</math>

Before physicists discovered the nature of electrical charge and electrical current, there wasn’t a way to identify whether the charge carriers were positively or negatively charged. Therefore, they made an assumption that it was positive charge that moves . But it turned out that their guess was wrong. We now know, it is the negatively charged electrons that provide electrical conductivity in metals or materials. Thus, in many of the conventions, physicists traditionally observe how the positive charge moves. Whereas chemists look at the displacement of an electron. As a result, the dipole moment can be written as going from left to right if you have a donor-acceptor molecule.

Suppose a quasi one dimensional D-conjugated bridge -A molecule with z the long axis and then apply an external field along z.

:<math>\overrightarrow{\mu}^\circ\,\!</math>
:<math>\rightarrow\,\!</math>

D ----- conjugate -------- A

:<math>\rightarrow\,\!</math> :<math>\overrightarrow{F}_x\,\!</math>
:<math>\leftarrow\,\!</math>

It can also be written, as going from right to left. Suppose that we have a donor acceptor molecule with a conjugated bridge between the two. In linear quasi- 1-demensional type molecules, the whole optical or non-linear optical responses will occur along the axis Z of the molecule.

=== Stabilization ===

 
[[Image:Stabilization.png|thumb|300px|]]

Assume you have molecule that has a positive pole and a negative pole. You can place an electric field along the main axis in two directions. At the first order you will only observe the permanent dipole moment of the molecule and its interaction of the field; thus you have a permanent dipole moment period. You have a plus and a minus. It is also important to know which situation, the one on the top or the one on the bottom, will be more stable. As a matter of fact, the one at the bottom will be the most stable situation. This is because when you have two dipole moments on top of one another, the anti-parallel? situation will be much more favorable then the parallel situation. In anti-parallel situation the positive charge is stabilized by the negative pole of the electric field and the negative charge is stabilized by the positive pole. Where as in the parallel situation there is a destabilization. Therefore, independently from the conventions in terms of the electric field and the dipole moment, it is clear which situation will lead to a net stabilization of the energy of the system and which one will lead to a destabilization.

At first order, nothing changes within the molecule.

At second order you will have a flux of electrons towards the left to counteract the external field in the lower case, and in the upper case you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself. At third order, it gets more complicated.

'''First order energy term'''
:<math>E(\overrightarrow{F}) - E^\circ = - \overrightarrow{\mu^\circ} \overrightarrow{F}\,\!</math>

:<math>= - \overrightarrow{\mu}_z ^{ \circ}- \overrightarrow{F}_z\,\!</math>

There is stabilization if :<math>\overrightarrow{F}_z\,\!</math> is parallel to :<math>\overrightarrow{\mu}^\circ\,\!</math>

There is destabilization if :<math>\overrightarrow{F}_z\,\!</math> is anti-parallel to :<math>\overrightarrow{\mu}_z^{\circ}\,\!</math>

In other words, with the indicated conventions, stabilization will occur if the field is parallel to the dipole moment and destabilization will occur in the opposing case. But again, that depends on the conventions chosen for the field and dipole moment. Remember that at first order in the field ( the linear term of the energy expression) only the interaction between the permanent dipole moment, is examined. However, at higher orders, we examine how the system responds to the external field on the molecule. As a result, we look at the polarizabiliy, or in the case of alpha the linear polarizability. In perturbation theory the second order term gives the stabilization.

This can easily be seen from the previous expression. Alpha is a summation over all excited states of the squares of the transition dipole, which makes it positive. The transition energy going from the ground state will always be positive by definition of the ground state.

'''Second order energy term'''
:<math>- \frac{1}{2} \alpha_{zz} \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

Alpha is positive and we multiply by F times F, which will be positive. Therefore, the whole second order term leads to a stabilization of the system.

Think back about the simple example shown previously. At first order, nothing changes within the molecule. At second order, look at the response of the molecule to the external field. What will happen here? What will happen is that you will have a flux of electrons towards the left to counteract the external field, and here you will have a flux of electrons toward the right. Thus, the system will always respond in a way to stabilize itself.

Alpha is a tensor of rank two and there are nine tensor components for alpha. Beta is a tensor of rank three. Since each of these indices can be x y z, there will be a possible of 27 tensor components.

'''third order energy term'''
:<math>- \frac{1}{6} \beta_{zzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\,\!</math>

There is stablilization or destablization depending on whether :<math>\overrightarrow{F}_z\,\!</math> is parallel or antiparallel to the vectorial part of β, and depends on the sign of β which depends on Δ μ eg

In a two state model expression, β depends very much on the difference in state dipole moment between the ground state and the active excited state. Β will be positive if that active excited state has a dipole moment that is larger than the dipole moment in the ground state, and β will be negative if the dipole moment in the excited state is smaller than in the ground state. This is an easy way of understanding the variation in the sign of β.

'''Fourth order energy term'''

:<math>- \frac{1}{24} \gamma_{zzzz} \overrightarrow{F}_z \overrightarrow{F}_z \overrightarrow{F}_z\overrightarrow{F}_z\,\!</math>

This is stabilizing if γ is >0 and destabilizing if γ is <0

For fourth order, in this case, the field components would lead to a positive term by necessity. Thus, we will have either stabilization if γ is positive or destabilization if γ is negative. This is consistent with the process called the self-focusing of light in the material with positive γ. If you shine a high intensity laser light into a molecule that has a very large positive γ response, the beam will self-focus. The system tends to go to higher local fields and therefore obtain a larger stabilization by focusing the light. Where as a negative γ leads to a defocusing of your light. This property can be use for protection from high intensity light.

Gamma is a tensor of rank four and there will be 81 tensor components. When looking at extended π conjugated molecules, (quasi 1-dimensional) the components along the long axis of the molecule will dominate everything. However, with molecules that become more complex in shape there are a number of components that can become important as well. Also, there are symmetry relationships among those components. In the literature on non-linear optics, there is something referred to as Climan symmetry that is based on the point groups of the different molecules that gives the relationship between the different tensor components. However, here we are mostly concerned with at the αzz component, the βzzz, or γzzzz What will be provided is a difference between the global value and the tensor component along the main axis. It is difficult to know whether the third order term leads to stabilization or destabilization because β could be positive or negative. Also, the combination of the three field terms can be positive or negative so it really depends.

=== Dipole changes ===
We can also look at what happens to the dipole moment. In the case of the α, the permanent dipole can be zero if we have a centrosymmetric molecule or it can be any value depending on the nature of the molecule. If it is non-centrosymmetric there will be an increase or decrease depending on the direction of the field at first order.

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} = \overrightarrow{\mu^\circ_z} + \alpha \overrightarrow{F_z}\,\!</math>

First order: The dipole increases or decreases according to whether Fz is parallel or antiparallel to :<math>\overrightarrow{\mu^\circ_z}\,\!</math>

Second order:

The change depends on how the field is aligned with respect to the permanent dipole moment. At the next order FF is always positive so dipole it will be decided by the value of β. The sign of β can often be related to the difference in dipole moment between the ground state and the active excited state. If there is an increase in the dipole moment going from the ground state to the excited state, β will be positive. That excited state now contributes to the description of the system because with a larger dipole moment, it is reasonable to assume that the μ of the system will increase. The opposite will occur for a negative β. All these considerations will become clearer when the perturbative expressions for β and γ are discussed in detail.

:<math>1/2 \Beta : \overrightarrow{F} \overrightarrow{F}\,\!</math>

In the context of the two state model, β has a sign of :<math>\Delta \mu_{eg}\,\!</math>

:<math>\beta >0 : \mu \uparrow\,\!</math>

:<math>\beta <0 : \mu \downarrow\,\!</math>

Third order
For the impact of γ, the dipole moment depends on the sign of γ and the field alignments in the expression of the dipole moment. There will be three fields that will play a role.

=== Calculation of polarizabilities ===
Polarization of a medium due to an electric field.

In spite of the different conventions used to look at the physics of the system, it is good enough to just look at what the external field with respect to the permanent dipole moment does. Papers in the field of non-linear optics, especially for inorganic materials, often look at the macroscopic polarization that occurs when the field is applied. Since the experimentalists are not concerned with the possible derivations that are necessary when calculating α, β, γ, they often use an expression that is a power series expansion instead of a Taylor series expansion.

This expression of the polarization of the medium corresponding to the possible permanent polarization when the material is non-central symmetric. The expression contains a first order term which is the first order electrical susceptibility. Remember, the :<math>\chi^{(1)}\,\!</math> is a tensor; there will be 9 tensor components there. That is the equivalent of α for the microscopic scale.

:<math>\overrightarrow{P} = \overrightarrow{P_0} + \chi^{(1)} \overrightarrow{F} + \chi^{(2)} \overrightarrow{F}\overrightarrow{F} +\ chi^{(3)} \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} +\,\!</math>

:<math>\chi^{(1)}\,\!</math> is the first order electrical susceptibility (second rank tensor).

:<math>\chi^{(2)}\,\!</math> is the first order electrical susceptibility (third-rank tensor).

:<math>\chi^{(3)}\,\!</math> is the third order electrical susceptibility, and so on.

Only :<math>\chi^{(1)}, \chi^{(2)}, \chi^{(3)}\,\!</math> will be considered, although experimentally there are people that have shown :<math>\chi^{(5)}, \chi^{(6)}\,\!</math> processes that are very specific.

Molecular materials at the microscopic level

:<math>\overrightarrow{\mu} = \overrightarrow{\mu_0} + \alpha \overrightarrow{F} + \beta \overrightarrow{F} \overrightarrow{F} + \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

where

:<math>\alpha\,\!</math> is first order polarizability

:<math>\beta\,\!</math> is the secord order polarizability or first order hyperpolarizability

:<math>\gamma\,\!</math> is the third order polarizability or second order hyperpolarizability

This is the corresponding expression for the dipole moment of a given molecule on the microscopic level. It is expressed in the power series expression. α is referred to as the polarizability. In the context of non linear optics, when looking at the β and γ terms, α can be more rigorously referred to as the first order polarizability. β is the second order polarizability or (some people prefer to use the expression) first order hyperpolarizability. γ is the third order polarizability or the second order hyperpolarizability. The reason why μ is expressed in both a power series expression and in a Taylor series expression is that most of the programs that make calculations use Taylor series expansion. However, the β or the γ that one calculates can differ from one program to another. It can differ by a factor of 2 for β, and by a factor of 6 for γ. Therefore, it is wise to also compare your calculated data with what is reported experimentally. Usually, experimentalists use a power series expansion. Thus, if they had done the calculation taking into account the Taylor series expansion, they will have immediately a difference by a factor of 2 or a factor of 6 with the experiment.

'''Stark Energy'''

Switching back to the Taylor series expressions. This shows the stark energy expression written in a more rigorous way taking into account for all the possible components of the field and for the tensor components of the α, β, and γ tensors.

:<math>E(F) = E_0 - \sum_{i} \mu_{0i}F_i - \frac {1} {2!} \sum_{ij} \alpha_{ij} F_i F_j - \frac {1} {3!} \sum_{ijk} F_i F_jF_k - \frac {1}{4!} \sum_{ijkl} \gamma_{ijkl} F_i F_j F_k\,\!</math>

'''Dipole Moment'''

This shows a similar expression for the dipole moment. These two expressions are fully consistent with each other, given the Hellman-Feynman theory.

:<math>\mu_i(F) = \mu_{0i} + \sum_j \alpha_ij F_j + \frac {1} {2!} \sum_{jk} \beta_{ijk} F_j F_k + \frac {1} {3!} \sum_{jkl} \gamma_{ijkl} F_j F_k\,\!</math>

:<math>\mu_i = - \frac {\partial E(f)}{ \partial F_i}\,\!</math>

These expressions clearly show what was confirmed earlier regarding the tensors and its respective rank. For example, γ will be a tensor of rank 4 because you are looking at the impact on the i component of the dipole moment when applying a field along j, a field along k, or a field along L. That is the reason why the α, β, and γ contain all those tensor components.

:<math>\alpha_{ij} = -\frac{ \partial ^2E(F)} {\partial F_i \partial F_j} = \frac {\partial ^1 \mu_i} {\partial F_j}\,\!</math>

:<math>\beta_{ijk} = -\frac{ \partial ^3E(F)} {\partial F_i \partial F_j \partial F_k} = \frac {\partial ^2 \mu_i} {\partial F_j \partial F_k}\,\!</math>

:<math>\gamma_{ijkl} = -\frac{ \partial ^4E(F)} {\partial F_i \partial F_j \partial F_k \partial F_l} = \frac {\partial ^3 \mu_i} {\partial F_j \partial F_k \partial F_l}\,\!</math>

=== Derivative Techniques ===

From those derivative expressions and the perturbative expressions, two types of calculations can be derived to evaluate the molecular polarizabilities from quantum-mechanical approaches. There is one major set of calculations that involve the derivation of either the energy or the dipole moment with respect to the external field. Those derivations can be done either numerically using methods referred to as finite-field methods, or analytically using Coupled Perturbed Hartree-Fock (CPHF) methods.

In a finite-field calculation, you take the interaction with the external field and put it into your Hamiltonian for the isolated molecule without any external perturbation. It has a kinetic term, a nucleic attraction term, a coulomb term, and exchange term. Now here, a fifth term is added to those present four terms. The fifth term expresses the interaction with your field. Several calculations are then made in which several values of the external field are taken into account. Then you do a numerical derivation of the dipole moments that you will have calculated as a response to the external field.

:<math>H(\overrightarrow{F} = H_0 - \overrightarrow{\mu} \overrightarrow{F}\,\!</math>

The MO's are self-consistant with the eigenfunctions of :<math>H(\overrightarrow{F})\,\!</math>. What is interesting with those finite field methods is that since the perturbation interaction with the electric field is put into the Hamiltonian, the molecular orbitals that are derived are affected by that interaction. Then the α, β, γ tensor components are calculated by applying standard numerical procedures. Calculations are made with different values of the field. Different values of for dipole moment for the molecule are obtained. A numerical derivation is then made to get to α, β, and γ. For instance, two calculations are made for the α ii component. The calculation is made with the field in one direction, and then again with the field in the opposite direction. It is important to have a value of the field that is large enough so that the molecule can respond and give a numerically accurate variation in the dipole moment. However, it should not be too large or the equivalent of a dielectric breakdown of your molecule will be obtained and the calculation will simply not converge. Therefore, it is crucial know what values of the fields are needed to evaluate.

:<math>\alpha_ii = \frac {\partial\mu_i} {\partial F_i} = \frac {1}{2F_i} [\mu_i(F_i) - \mu_i(-F_i)]\,\!</math>
:<math>
\gamma_{iiii} = \frac {\partial^3 \mu_i} {\partial F_i \partial F_i \partial F_i} = \frac {1} {48F_i^3} [\mu_i(3F_i)-\mu_i(-3F_i)- 3\mu_i (F_i)+ 3\mu_i (-F_i)]\,\!</math>

We pick a compromise value that is able to insure accuracy but also avoid divergence.

:<math>F_i \approx 5 x 10^8 Vm^{-1}\,\!</math>

'''Coupled perturbed Hartree-Fock method'''

Another method that can be used to make those calculations is the analytical methods with analytical expressions for the variation of the energy with respect to the electric field.

:<math>\beta_{ijk} = - \frac {\partial^3 E(F)} {\partial F_i \partial F_j \partial F_k}\,\!</math>

=== Perturbation techniques ===

'''Sum over state (SOS) method'''
Besides making numerical or analytical calculations based on the derivation expressions, the perturbation theory expressions can also be used. This method is usually referred to as Sum Over States (SOS) method. This method was seen before for α. It is based on the perturbation expression for Stark energy terms which are related to optical nonlinearities based on their order in the field strength. α is calculated by evaluating the transition dipoles and the transition energies for all the excited states in the molecule.

You can look at the convergence of your values as a function of going over many excited states. However, it is important to understand that the higher energy you go, the larger the denominator becomes. Therefore, those terms will have smaller weight. Also, at very high excited states, the transition dipole will die down as well. For example, in the case of α the lowest excited states have the largest response.

:<math>\alpha_{ij} = \sum_m \frac {<\psi_0|\mu_i | \psi_m > <\psi_m|\mu_j | \psi_0 >} {E_m- E_0}\,\!</math>

For the β terms, we have exactly the same components. However, the expression looks more complicated because it contains a double summation over excited states. That is transition dipole going from the ground state n to excited state to m. Then it goes from excited state m back to the ground state. The denominator has the transition energies. There is also a second term that goes over a summation over excited state due to the dipole moment starting in the ground state. Then there is a transition dipole going from the ground state to excited state n and then it comes back from n to the ground state, over transition energies. To generate these expressions go through the perturbation theory and work the second order and the third order perturbation theory expression, one can do so by placing the dipole (er), the dipole operator, and the electric field.

:<math>\beta_{ijk} = \sum_n \sum_m \frac {<\psi_0|\mu_i | \psi_n > <\psi_m|\mu_j | \psi_0 ><\psi_m|\mu_k | \psi_0 >} {(E_n- E_0)(E_m- E_0)} - \sum_n \frac {<\psi_0|\mu_i | \psi_0 > <\psi_0|\mu_j | \psi_n ><\psi_n|\mu_k | \psi_0 >}{(E_m- E_0)(E_n- E_0)}\,\!</math>

The reason why it is interesting to evaluate the non-linear optic properties with those perturbation expressions (sum over state expressions) is because it pinpoints which excited states play important roles in optical and non-linear optical response. Another reason why they are heavily exploited is because the frequency dependence can easily be introduced with the response. With the finite-field method, it gets extremely complicated to introduce the frequency dependence.

The finite-field method is usually incorporated in many quantum chemistry packages. You just press key telling “calculate the molecular polarizabilites” and then you get numbers for those polarizabilites. However that is the problem; it only gives numbers and these are the static values where ω is equal to zero.
It doesn’t provide an in depth understanding of what is occurring. There have been extensions to those methods that provide some kind of understanding regarding finite field methods of the local spatial contributions to the non-linear optical response. But it is a sophisticated approach that is not often used. Thus, in many instances, it more beneficial to use the sum-over states expression because it gives an idea of which electronic states matter. Also, frequency dependence can easily be introduced. The same thing can be done at the γ level.

:<math>\beta^{SHG}(-2\omega;\omega \omega) = 1/2 \sum_n \sum \_m \frac {<\psi_0 | \mu_i | \psi_n><\psi_n|\mu_j|\psi_m><\psi_m |\mu_k|\psi_0>} {(\hbar\omega-(E_m-E_0))(2\hbar\omega-(E_m-E_0))}\,\!</math>

where

:<math>(\hbar\omega-(E_m-E_0))\,\!</math> represents one-photon resonance

:<math>(2\hbar\omega-(E_m-E_0))\,\!</math> represents two-photon resonance

Useful simplifications can be introduced if it is found that a single excited state dominates second order polarizability.

 
[[Image:Perturblevels2.png|thumb|300px|]]
In the case of the first term, when the excited state m is different from excited state n, it will go from the ground state to m and then to n,. Then from n back down to the ground state. This slide shows the pictorial description of the products of the transition dipoles. However, if m is equal to n, you will go from the ground state to m, but then stay on m. Then you will come back down to the ground state. Staying on m if m is equal to n simply means that the state dipole moment of excited state m is being observed. Then there is a second term here that comes with a minus contribution where you have the state dipole in the ground state, and then you go from the ground state to m and then from m back to the ground state. This is the pictorial way that can describe the 3 different types of terms is found in the sum over states expression.

It is possible to take this expression, show that everything simplifies when approximating that there is a single excited state e that matters.If there is a single excited state e that provides the most significant contribution to the second order in non-linear optical response of the system, there are simplifications that can be obtained.

If one excited state e is dominant:

:<math>\beta \approx \frac {<g|\mu|e> <e|\mu|e> <e|\mu|g>} {(E_e-E_g)}^2 - \frac {<g|\mu|g> <g|\mu|e> <e|\mu|g>} {(E_e-E_g)}\,\!</math>

:<math>\approx \frac {|\mu_{eg}|^2 \Delta \mu_{eg}} {|\hbar \omega_{eg}|^2}\,\!</math>

The sign of beta depends on the sign of Δμ. If the dipole moment in the excited state is larger than the dipole moment in the ground state then you will have a positive β. If the charge transfer is less in the excited state then you have a negative β. A molecule with a large with a large dipole in the ground state such as a zwitterion will have a smaller dipole moment in the excited state.

This is known as a two state model <ref>J.L Oudar, J. Chem. Phys. 67, 446 (1977)</ref> which guided chromophore development for the last 30 years until we the theory of bond length alternation gave us better insight. From this simple expression, it became easier to know whether one needed a larger oscillator strength or a system that has a very large acentric coefficient as one goes from the ground state to an excited state. Most importantly, what one needs is a difference in dipole moment as large as possible between the ground state to the excited state. This makes sense because if one started with a dipole moment that is not very large in the ground state and in the excited state, one would generate a very large dipole moment, which indicates a separation of charges and a highly polarizable system.

Also, it is important to have small transition energies depending on the process that is being observed. For instance, in the early 90’s, there great interest in second harmonic generation in which the input frequency is doubled when output. This process was used for converting a red laser into a blue laser which was important until people developed lasers that could shine without doubling the frequency. In order to get a blue beam as an output when the red laser serves as your input, we must prevent the blue beam from being absorbed by that material which provides for the second harmonic generation. Therefore, its works best when that material is basically transparent. On the other hand, for electro optics applications, the input and output frequencies are the same. This allows one to deal with materials that have a much smaller band gap, especially if the input frequency is in the IR.

 
=== Sum over states expression for γ ===
[[Image:energylevels-nmpg.png|thumb|300px| ]]

The full expression for γ can be simplified down to a three term model. It is dependent on the input frequencies ω1ω2ω3.

:<math>\Gamma_{abcd} (-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) = \hbar^{-3} K(-\omega_\sigma; \omega_1, \omega_2, \omega_3 ) x</math>

:<math>\lbrace \rbrace</math>

:<math>x \left\lbrace + \sum_p \left[ \sum_{m,n,p} \frac {<g | \mu_a |m >
<m | \overline{\mu_b} |n ><n | \overline{\mu_c} |p >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - \omega_2- i \Gamma_{ng})(\omega_pg - \omega_2- i \Gamma_{pg})} \right] -\sum_p \left[ \sum_{m,n} \frac {<g | \mu_a |m >
<m | \mu_b |g ><g | \mu_c |n ><n | \mu_d |g >} {(\omega_mg - \omega_\sigma - i \Gamma_{mg})(\omega_ng - \omega_1 - i \Gamma_{ng})(\omega_ng - \omega_2- i \Gamma_{ng})} \right] \right\rbrace\,\!</math>

Note that:

:<math><m | \overline{\mu_b} |n > = <m | \mu_b |n > - \delta_{mn} <g | \mu_b |g ></math>

In the numerator there is summation over four transition dipole moments (energies) and in the denominator summation over 3 excited states. There can be resonance and the photons are absorbed. Gamma is related to the lifetime of the excited state. According to the Heisenberg principle if the excited state lives for a very short time then the energy will less precise.

At the static limit where frequency goes to zero the expression becomes simpler.

:<math>\gamma \propto \sum_{m,n,p \neq g} \frac {<g|\mu_z|m ><m|\overline{\mu_z}|n> <n|\overline{\mu_z}|p><p|\mu_x|g>}{E_{gm} E_{gn} E_{gp}} - \sum_{m,n \neq g} \frac {<g|\mu_z|m><m|\mu_z|g><g|\mu_z|n><n|\mu_z|g>}{E_{gm} E^2_{gn}}</math>

for <math>\omega \rightarrow 0</math>

Note that:

:<math><m | \overline{\mu_z} |n > = <m | \mu_z |n > - \delta_{mn} <g | \mu_z |g ></math>

 
'''Third order expression'''

If you make the assumption that the ground state g is only strongly coupled to a single excited state e which is in turn coupled to other states excited states e’, there is a three term expression. The positive expression then generates two terms depending whether e’ is different from e yielding the first expression, or e’ is different from e producing the Δ E expression.

:<math>\gamma_{zzzz} \propto \sum_{e\prime} \frac {M_{ge}^2 Me_{e\prime}^2} {E_{ge}^2 E_{ge\prime}} - \frac {M_{ge}^4} {E_{ge}^3} + \frac {M_{ge}^2 \Delta_{\mu_{ge}} ^2} {E_{ge} ^3}</math>

The last term exists only in noncentrosymmetric molecules. All the terms are positive because of squared terms and therefore the first and last term increase γ while the middle term decreases γ. The permanent dipole moment is zero in centrosymmetric molecules and β is also zero. α and γ can have non-zero values regardless of the symmetry of the molecule.

=== Calculation of alpha components ===

α has a simple sum over states expression given a single excited state that is strongly coupled to the ground state. This the square of the transition dipoles over the transition energy. α is always positive, whereas β and γ can be positive or negative. α always provides a stabilization.

:<math>\alpha \approx \frac {<g|\mu|e> <e|\mu |g>} { \hbar \omega_{eg}}</math>
 
The first excited state represents an homo to lumo promotion, to the 1bu state.

[[Image:Homotolumo.png|thumb|300px|Homo to lumo promotion to first excited state]]
 
'''Simple conjugated chains, Polyenes (polyacetylenes)'''
[[Image:Polyeneshift.png|thumb|300px| ]]
In polyenes as you move from g to e the π electronic cloud is shifted from one bond to the next. The opposite transition also occurs creating a resonance structure. Alphazz (zz is the long axis tensor) is approx n^1.6 in polyenes. Aromaticity is not detrimental to alpha.
 
[[Image:Alphaperbond.png|thumb|300px|Alpha per bond according to polyene length. From Bodart 1985<ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref> ]]
As polyene chains get longer there are more electrons therefore larger polarizability. Up to 10 double bonds α increases but then saturation occurs after some 15-20 double bongs (~50 A). It is useful normalize this measurement by considering the polarizability per double bond.

The evolution of alphazz as a function of chain length L:
:<math>\alpha_{zz} \sim L^{1.6}</math>

other theories predict a much stronger evolution with length:

:<math>\alpha_{zz} \sim L^{3}</math>

:<math>\gamma_{zz} \sim L^{5}</math>

So one molecule with 30 double bonds (past saturation) will have the same polarizability as two molecules with 15 bonds. There is no advantage in polymers longer than saturation.

The polarizability strongly increases as the degree of bond-length alternation goes down and you approach the cyanine limit. This suggests a larger polarizability in the presence of conformational defects such as solitons and polarons. Thus BLA can be used to influence the polarizability. <ref>de Melo and Silbey, J. Chem. Phys. 88, 25588 (1988)</ref>

[[Image:Alphaperbond_bla.png|thumb|300px|Alpha evolution by chain length for various bond length alternations <ref>Bodart et al Cand. J. Chem. 63, 1631(1985)</ref>]]

 

=== Calculation of β components ===
[[Image:Betameasured.png|thumb|300px| ]]
A number of benzene and stilbene were studied by Lak Tak Cheng at Du Pont<ref>EXPERIMENTAL INVESTIGATIONS OF ORGANIC MOLECULAR NONLINEAR OPTICAL POLARIZABILITIES .1. METHODS AND RESULTS ON BENZENE AND STILBENE DERIVATIVESCHENG, LT; TAM, W; STEVENSON, SH; MEREDITH, GR; RIKKEN, G; MARDER, SRJOURNAL OF PHYSICAL CHEMISTRY 95 (26):10631-10643 1991</ref>. beta changed dramatically depending on the moieties involved. As you increase the length of the conjugated bridge you increase the electrons, and the system has the capability of separating the charges over a long distance. Vinylene moity causes a large beta response. So this is an indication that vinylene has a large effect on higher order polarizabilities beta and gamma. Decreasing the aromaticity with a thiophene ring increases the delocalization and increases the response further.

On the basis of the two state model a non centrosymmetric molecule can be polarizable. Molecules with a large dipole along the long axis will orient in an antiparallel manner in a crystal or thin film. This means that even though the molecule has a high β the χ(2) will be small for the whole material. The concept of crystal engineering is promote noncentrosymmetry at the macroscopic scale.

[[Image:Pushpull.png|thumb|300px| ]]
 
One approach is to minimize the dipole moment μg in the ground state.

[[Image:Triaminotrinitrobenze.png|thumb|300px|Triaminotrinitrobenzene is nonpolar but has a beta]]

This substance triamino trinitrobenzene is a noncentrosymmetric structure but the local dipole moments add up to zero in ground state<ref>J.Zyss, Nonlinear Optics Quantum Optics 1, 3 (1991)</ref>. Because dipole moment is zero it will crystalize in a noncentrosymmetric manner. Like boron trifluoride has 3 very polar groups but the geometry of the molecule cancels out the polarity.

The beta for triamino trinitrobenzene is not zero becuase beta has components from octopoles as well as dipoles. In this case the two state model breaks down.
 
Chain length dependence of static χ(3) for polyacetylene. Static χ(3) starts to saturate at 50-60 carmbons <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref> The graph shows a sigmoid shape with a slow start, a rapid increase and then a leveling off. This is common in these systems.

For short chains (~10 carbons):
:<math>\gamma(0) \sim N^{4.3}\,\!</math>

[[Image:Chainlength_chi3.png|thumb|300px| SSH/SOS calculation for χ (3) versus carbon chain length <ref>Shuai et al., Phys. Rev. B 44, 5962 (1991)</ref>]]

For long chains the separation for γ occurs at a much longer distance than for α. For gamma there the contribution from the first excited state and there is a higher lying excited state that further adds to polarizability.

The first data from free electron laser revealed trends in χ (3)There is a frequency dependence in chi 3 caused by resonances. This causes spikes in the chi (3) at three photon resonance at .6ev (3 x .6 is 1.8, the bandgap for transpolyacetylene) and .9 spike from two photon resonance.
[[Image:Electrontransfer_polyacetylene.png|thumb|300px|Free electron transfer data for trans-polyacetylene <ref>Fann et al. PRL 62, 1492 (1989)</ref> ]]

 
=== Calculation of gamma ===
We can calculate the evolution of gamma in oligothiophene with increasing number of rings.
[[Image:Gamma_oligothiophenes.png|thumb|300px| Gamma versus number of rings for oligothiophenes INDO- MRDCI SOS]]
After 6 or 7 rings there is a strong increase. This has been confirmed experimentally <ref>Thienpont et. al. PRL 65, 2140 (1990)</ref> The gammas for polythiophenes are about one order of magnitude than the polyenes.
 

'''polyarylene vinylene chains'''
[[Image:Polarylenevinylene.png|thumb|300px|γ values for various oligomers with 30 π electrons]]
Lower aromaticity of the oligomers with 30 pi electrons increases gamma. <ref>Phys. Rev. B 46, 4395 (1992)</ref>

Polythienylene vinylenes have higher responses than polyphenylene vinylenes and polythiophenes.

Polyenes are extremely polarizable structures.

The following relation regarding bandgap (Eg) should be taken with caution because of the influence of molecular structure

:<math>\gamma \propto (1/Eg)^6\,\!</math>

Third order polarizabilities are significantly larger in polyarylene vinylenes than polarylenes.
 
Quinoid structures are aromatic and have lower gamma, so consider semiquinoid structures.

[[Image:Quionoid.png|thumb|300px| ]]
 
Be careful of using scaling laws going from alpha to gamma, they are not proportional.

[[Image:Scaling.png|thumb|300px| ]]

 

'''Fullerenes C60 and C70'''
α is typically measure in 10-24, β in 10-30 and γ in 10-36 esu units, losing 6 orders of magnitude with each level. This is why powerful lasers are needed to observe non-linear optical phenomena. In moving from fullerenes to the corresponding polyene there is an increase of 2 orders of magnitude. The more stretched out molecule is more polarizable than the more collected arrangement of C60

Static value in 10-36esu
{| {{prettytable}}
|-
! gamma
! C60
! C70
! C60 H62
|-
| γzzzz
| 226.1
| 816.5
| 4x105
|-
| γzzyy
| 62.5
| 141.1
| 627
|-
| γyyyy
| 212.8
| 516.3
|-
| γxxxx
| 212.8
| 516.3
|-
| γ
| 202.8
| 862
| 8x104
|}

'''Second harmonic generation in C60'''
There is an evolution of an electric quadrupole and magnetic dipole which contributes to the second harmonic generation in SOS/VEH calculations. So you need to go beyond only the electric dipole and include the magnetic dipole.

{| {{prettytable}}
|-
! hω
! exp SHG
! calculated MD
! SOS/VEH EQ
|-
| 1.2 eV
| 2.1 x 10-9 esu <ref>Koopmans et al PRL 71,3569(1993)</ref>
| 0.9 x 10-9 esu
| 0.2 x 10-9 esu
|-
| 1.8 eV
| 1.8 x 10-9 esu <ref>Wang et al. Appl.Phys. Lett. 60, 810 (1991)</ref>
| 1.0 x 10-9 esu
| 0.3 x 10-9 esu
|}

== References ==
<references/>

Quantum-Mechanical Theory of Molecular Polarizabilities

2011-06-28T21:26:58Z

Chrisb:

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Quantum-Mechanical Theory of Molecular Polarizabilities up to Third Order

The literature refers to first order, second order and third order polarizabilities. You may also see “first order hyperpolarizability “ which is the same thing as second order polarizability. These conventions may be confusing.

Our goal is to be able to relate the chemical structure and the nature of the pi-conjugated backbone, and the nature of donor and acceptors to the to kind of non-linear response that can be measured or calculated.

The expression for dipole moment is simply the sum of all the point charges over all those point charges of the charge itself times the position of that charge.

=== Dipole moment ===

 [[Image:Dipole_example.png|thumb|300px|]]
The drawing of the charges shows a point charge of +1 on the left side and a point charge of -1 on the right side. The origin of the system of coordinates is in the middle of these charges. The point charge of +1 is -5 angstrom away from the origin and the point charge of -1 is +5 Ångström away on the x-axis. The first step to calculate the dipole moment is to multiply the charge +1 by the position -5 Angstrom for the +1 point charge, which will be -5eA (electron times angstrom). Then do the same for the -1 point charge (-1 * 5) which equals -5eA as well. Finally the sum of these values (-5eA + -5eA) -10eA, which corresponds to about -48 debyes, will be the dipole moment. The literature does not use the SI units of the dipole moment because the SI units use coulomb for the charge and meters for the distance which are macroscopic units. Instead, the dipole moment units use electron units and Angstrom (eA), or Debye.

:<math>Cb \cdot \, m \, or Debye \approx 4.8 q(\vert e \vert) * d( A )\,\!</math>

If there is a donor acceptor compound, a full charge transfer can occur. The drawing shows the two point charges that are 10 angstrom apart with the length of the conjugated bridge between them. Then with one full electron charge transfer across 10 angstrom, a dipole moment of 50 debyes will be obtained (this is quite large). If the donor and acceptor are about 10 angstrom apart and the dipole moment is about 10 debyes, the system has a charge transfer of about 0.2 electrons from the donor to the acceptor assuming it a simple dipole.

This example is a simplified model where the origin of the system of coordinates was placed midway between the two charges. However, what would happen if the origin of the coordinates is placed on the +1 point charge? What will be the dipole moment? To answer this question, the +1 point charge will be multiplied by a distance of 0 (due to the origin of the coordinates) and that will not contribute anything to the dipole moment. The -1 point charge is now at a distance of 10 angstroms away and is multiplied by a charge of -1 which is -10 eA or about -50 debyes (-48 precisely).
Consider the case where the molecule is not neutral but instead a cation is formed with a +2 charge on the left.
If the origin of the system of the coordinates is at the +2 point charge, the calculated dipole moment will be -10 eA. But if the origin of system coordinate is placed on the -1 point charge, the dipole moment will be -20 eA. An important lesson out of these examples is that with a charged system, the dipole moment depends the choice of origin of coordinates. The dipole moment is no longer perfectly defined.

In the previous example in which the point origin of coordinates was placed on the +1 point charge, there was a net charge so the monopole is not 0. Therefore, the dipole and all the other poles (quadrapole, octapol, hexapole etc.) depend on the origin of the system of coordinates. However, if the monopole is 0, then the dipole doesn’t depend on the origin of coordinates. But if the dipole is not 0, which means there is a permanent dipole moment in the molecule, then the dipole and the other poles will depend on the origin of coordinates. If the monopole is 0 and the dipole is 0, due to the central symmetric molecule, then the quadrapole will not depend on the choice of the origin of the system of coordinates. It is only when the molecule doesn’t carry a net charge that the dipole is really defined. Otherwise, if the molecule does carry a net charge, the dipole is not defined and it will depend on the origin of the system of coordinates that is chosen. This also creates further complications as it is important to make sure that the point of references is the same for similar systems in order to make comparisons. But for now, when one talks about dipoles, the dipoles will be from neutral molecules so that there will be no complications. In the case of two photon absorption molecules the quadrupole moments can be important. But since these molecules are usually centro symmetric and the dipole is 0, the quadrapole will not depend on the choice of the origin of the system of coordinates.

In the case of point charges where it is easy to describe the degree of dependencies or on the system of coordinates.

:<math>\overrightarrow{\mu} = \sum_i q_i \overrightarrow{r_i}\,\!</math> is the dipole for point charges

But if we have a molecule, the Heisenberg principle does not allow a precise location of the electron. Therefore, wave functions are needed. To get the dipole moment, it is necessary to look at the electronic densities at a given point in space. The expression consists of a wave function at a given point in space r (in other words, wave function at r) and the dipole operator. The dipole operator is the electronic charge of the particles being discussed multiplied by the position of those charges. Since r doesn’t modify Ψ, the Ψ can be arranged so that a psi squared is obtained, which is an electronic density. This makes sense because the wave function squared is an electronic density, and so the expression is equal to an electron density multiplied by a position. Those two expressions are perfectly consistent with one another.

:<math>\overrightarrow{\mu} = \int_{-\infty}^{\infty} \Psi^* e \overrightarrow{r} \Psi d \overrightarrow{r}\,\!</math> is the dipole for a molecule

where
:<math>\Psi^*\,\!</math> is the wave function for position r

:<math>e \overrightarrow{r}\,\!</math> is the dipole operator relating the charge and position of charges.

:<math>\langle \Psi \vert \overrightarrow{r} \vert \Psi \rangle\,\!</math> determines the dipole moment in the state described by the wave function Ψ

:<math>\overrightarrow{\mu}_g = \langle \Psi_g \vert \overrightarrow{r} \vert \Psi_g \rangle\,\!</math> This is just the Dirac expression for this integral expression.

The dipole moment can be examined at the excited state and this is critical for second order materials. To find the dipole moment in the excited state S1 of organic chromaphore, use the wave function of that particular excited state.

=== Transition Dipole ===
The dipole moment should not be confused with the transition dipole in which describes the transition from a given state to another state. Usually it is the transition from ground state to an excited state. In that case, the wave function psi g will be that of the initial state and the wave function psi e will be that of the final state.

:<math>\overrightarrow{\mu}_{eg} = \langle \Psi_e | e \overrightarrow {r} | \Psi)g \rangle\,\!</math> the transition dipole

:<math>\overrightarrow{\mu}_{eg} = \int \Psi^*_e e \overrightarrow{r} \Psi _g d\overrightarrow{r}\,\!</math>

Since r, the operator, multiplies but not modifies the wave function, the r term can be rearranged in this way.

:<math>\overrightarrow{\mu}_{eg} = \int \Psi^*_e (\overrightarrow{r}) \Psi _g (\overrightarrow{r}) e\overrightarrow{r} d\overrightarrow{r}\,\!</math>

These wave functions will be key to the description of polarizabilities in molecular compounds.
In this form the equation shows that in order to have large transition dipoles, it is necessary to have a reasonably good wave function overlap between the wave function of the initial state and the wave function of the final state. At the same time, it is also necessary to have that overlap for large values of r.

This means that if the two charges separate due to the polar attraction of the electric field of the light, it is clear that as the distance between two charges get larger, the dipole will become larger as well. Thus, the r value shows that dipole moment will be larger when the charges separate over a larger distance.

=== Polarizablity to polarization ===

Polarizability refers to a molecule and polarization refers to a material. When a molecule with a very large polarizability or material with a very large polarization is place in an electric field the electrons can move a large distance and delocalize a lot resulting in a separation of charges. Therefore, the larger the distance, the larger the state dipole, and the larger the transition dipole.

We will examine the behavior of individual molecule and then see what happens at the level of the material when many molecules come together.

'''Molecular Level'''

At the microscopic (molecular) level, a molecule that can have a permanent dipole moment , mu0 (permanent dipole) will have an induced dipole moment when it is exposed to electric field (or the electric field of the light). That induced dipole moment is proportional to the electric field an alpha (the polarizability) and the electric field. Thus, we can extend the analogy by acknowledging that the larger the alpha, the larger the polarizability, and the larger the induced dipole moment.

:<math>\overrightarrow{\mu} (induced) = \alpha \cdot \overrightarrow{E}\,\!</math>

where

:<math>\overrightarrow{\mu}\,\!</math> the dipole; a vector

:<math>\overrightarrow{E}\,\!</math> the electric field; a vector

:<math>\alpha\,\!</math> the polarizability is a tensor of rank 2

The electric field, the dipole moment, and induced dipole moment are vectors that have x, y, z components but alpha, the polarizability, is actually a tensor. It is a tensor because if an electric field is applied along x, not only can a dipole moment be generated along x but an induced dipole moment can also be generated along y or z. Therefore the induced dipole moment along axis i will depend on all the components of the field and these polarizability tensor components. So by measuring the induced dipole along mux, the x component of the field can also be observed and an alpha xx tensor component will be obtained.

:<math>\overrightarrow{\mu}_i = \sum_j \alpha_{ij} \overrightarrow{E}_j\,\!</math>

Those polarizability tensor components are a function of the wavelength.

:<math>\alpha_{ij} = f(\omega)\,\!</math>

Usually in the literature one talks about wavelengths and then shows the angular frequency, omega, not lambda. This may be confusing. The reason why the angular frequency is used is simply because the wavelength of the light refers to a very specific oscillation frequency of the electric field of the light.

'''Material Level'''

When looking at the macroscopic scale of the molecular material, one talks about the induced polarization of the material. Just like the microscopic scale, the induced polarization will be directly proportional to the electric field at the macroscopic scale.

:<math>\overrightarrow{P}(induced) = \Epsilon_0 \cdot \Chi_{el} \cdot \overrightarrow{E}\,\!</math>

where

:<math>\Epsilon_0\,\!</math> is the electric permittivity of a vacuum

:<math>\Chi_{el}\,\!</math> is the electric susceptibility of the material; a tensor

The displaced electric (D) field takes into account the external field, like the field due to the light and the polarization.

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} + \overrightarrow{P}\,\!</math>

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} + \Epsilon_0 \cdot \Chi_{el} \cdot \overrightarrow{E}\,\!</math>

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} (1+ \Chi)\,\!</math>

The displaced electric field can be expressed as the electric permittivity of the vacuum times the external field times (1 + χ). (1+χ) represents the electric susceptibility, which is an important characteristic of the material. It is also known as the dielectric constant. Dielectric constant is a function of frequency and wavelength and so it varies as a function of wavelength. Thus, it is constant to some extent. The dielectric constant is actually defined as the ratio of the electric permittivity of the medium itself over that of the vacuum.

:<math>\Epsilon_d = \frac {\Epsilon} {\Epsilon_0}\,\!</math> is the dialectric constant

Therefore, the dielectric constant has no dimension since it is a ratio of those two values. By substituting the ratio in the expression for the displaced electric field, a compact expression can be obtained that expresses the displaced electric field as simply the external field times the electric permittivity of the material.

:<math>\overrightarrow{D} = \Epsilon \overrightarrow{E}\,\!</math>

The permittivity and dielectric constant are a function of frequency (ω) or wavelenght (λ).

:<math>\Epsilon_{ij} (\omega) = 1 + \chi_{ij} (\omega)\,\!</math>

=== Relation between dielectric constant and refractive index ===

It is useful to keep in mind the simple relationship between the refractive index and the dielectric constant. Recall the definition of the speed of light, which is 1 over the square root of the electric permittivity of vacuum times the magnetic permeability of vacuum. When light enters a material (a medium), its speed becomes v, which is 1 over the square root of the product of the electric permittivity of that material or medium times the magnetic permittivity of that material or medium. In this case, mu here is the magnetic permittivity; not the dipole moment.

:<math>c = \frac { 1} {\sqrt{\Epsilon_0 \cdot \mu_o}}\,\!</math>

where
:<math>\Epsilon_0 \,\!</math>is the permittivity of vacuum

:<math>\mu_0 \,\!</math>is the magnetic constant or permeability of vacuum

The ratio of the speed of light in vacuum over that in the in the medium is the index of refraction.

:<math>\frac c v = n =\sqrt {\frac {\Epsilon \mu} {\Epsilon_0 {\mu_0}}}\,\!</math>

In the case of weakly magnetic materials μ = μ0

:<math>n(\omega)\approx \sqrt{\Epsilon_D (\omega)}\,\!</math>

The ratio of the speed of light in vacuum over that in the in the medium is the index of refraction.

:<math>n^2(\omega) \approx \Epsilon_D(\omega)\,\!</math>

For materials in which there is no magnetic activity, the magnetic permeability of the material is basically the magnetic permeability of the vacuum. Then the ratio of μ to μ0 will be close to one and thus the index of refraction will be nearly equal to the square root of the electric permittivity of the medium over that of the vacuum; this is the dielectric constant of the material. Often the interaction of the magnetic field with the light of the material is much smaller than the interaction of the electric field. Therefore, in many instances you can neglect the magnetic interactions and this expression would hold.

Eventually, this comes to an expression where the index of refraction squared corresponds to the dielectric constant. This is the connection between the index of refraction and the dielectric constant. It is important to remember that both the index of refraction and the dielectric constant depend on frequency. Also, often many people in the optics or photonics community casually quote the index of refraction for that material or the dielectric constant you will find that when the two numbers are compared, this expression appears to not hold true. This inconsistency is not due to an issue related to the magnetic material but it is simply because when people casually talk about the dielectric constant or index refraction, they are referring to the constant at very small frequency or in a static context where frequency tends to zero.

When people casually talk about the index refraction of the material, they usually mean optical frequencies. Optical frequencies are frequencies that correspond to the visible. Again, when one casually talks about index of refraction, one would imply optical frequencies where as in dielectric constants, one would imply very low frequencies.

Also, note that n, mu and Epsilon are also complex quantities because they may involve both dispersion effects and attenuation effects. As soon as there is absorption, then that becomes an imaginary component to the expression. Thus, all of those are complex numbers.

Quantum-Mechanical Theory of Molecular Polarizabilities

2011-06-28T21:26:34Z

Chrisb:

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Quantum-Mechanical Theory of Molecular Polarizabilities up to Third Order

The literature refers to first order, second order and third order polarizabilities. You may also see “first order hyperpolarizability “ which is the same thing as second order polarizability. These conventions may be confusing.

Our goal is to be able to relate the chemical structure and the nature of the pi-conjugated backbone, and the nature of donor and acceptors to the to kind of non-linear response that can be measured or calculated.

The expression for dipole moment is simply the sum of all the point charges over all those point charges of the charge itself times the position of that charge.

=== Dipole moment ===

 [[Image:Dipole_example.png|thumb|300px|]]
The drawing of the charges shows a point charge of +1 on the left side and a point charge of -1 on the right side. The origin of the system of coordinates is in the middle of these charges. The point charge of +1 is -5 angstrom away from the origin and the point charge of -1 is +5 Ångström away on the x-axis. The first step to calculate the dipole moment is to multiply the charge +1 by the position -5 Angstrom for the +1 point charge, which will be -5eA (electron times angstrom). Then do the same for the -1 point charge (-1 * 5) which equals -5eA as well. Finally the sum of these values (-5eA + -5eA) -10eA, which corresponds to about -48 debyes, will be the dipole moment. The literature does not use the SI units of the dipole moment because the SI units use coulomb for the charge and meters for the distance which are macroscopic units. Instead, the dipole moment units use electron units and Angstrom (eA), or Debye.

:<math>Cb \cdot \, m \, or Debye \approx 4.8 q(\vert e \vert) * d( A )\,\!</math>

If there is a donor acceptor compound, a full charge transfer can occur. The drawing shows the two point charges that are 10 angstrom apart with the length of the conjugated bridge between them. Then with one full electron charge transfer across 10 angstrom, a dipole moment of 50 debyes will be obtained (this is quite large). If the donor and acceptor are about 10 angstrom apart and the dipole moment is about 10 debyes, the system has a charge transfer of about 0.2 electrons from the donor to the acceptor assuming it a simple dipole.

This example is a simplified model where the origin of the system of coordinates was placed midway between the two charges. However, what would happen if the origin of the coordinates is placed on the +1 point charge? What will be the dipole moment? To answer this question, the +1 point charge will be multiplied by a distance of 0 (due to the origin of the coordinates) and that will not contribute anything to the dipole moment. The -1 point charge is now at a distance of 10 angstroms away and is multiplied by a charge of -1 which is -10 eA or about -50 debyes (-48 precisely).
Consider the case where the molecule is not neutral but instead a cation is formed with a +2 charge on the left.
If the origin of the system of the coordinates is at the +2 point charge, the calculated dipole moment will be -10 eA. But if the origin of system coordinate is placed on the -1 point charge, the dipole moment will be -20 eA. An important lesson out of these examples is that with a charged system, the dipole moment depends the choice of origin of coordinates. The dipole moment is no longer perfectly defined.

In the previous example in which the point origin of coordinates was placed on the +1 point charge, there was a net charge so the monopole is not 0. Therefore, the dipole and all the other poles (quadrapole, octapol, hexapole etc.) depend on the origin of the system of coordinates. However, if the monopole is 0, then the dipole doesn’t depend on the origin of coordinates. But if the dipole is not 0, which means there is a permanent dipole moment in the molecule, then the dipole and the other poles will depend on the origin of coordinates. If the monopole is 0 and the dipole is 0, due to the central symmetric molecule, then the quadrapole will not depend on the choice of the origin of the system of coordinates. It is only when the molecule doesn’t carry a net charge that the dipole is really defined. Otherwise, if the molecule does carry a net charge, the dipole is not defined and it will depend on the origin of the system of coordinates that is chosen. This also creates further complications as it is important to make sure that the point of references is the same for similar systems in order to make comparisons. But for now, when one talks about dipoles, the dipoles will be from neutral molecules so that there will be no complications. In the case of two photon absorption molecules the quadrupole moments can be important. But since these molecules are usually centro symmetric and the dipole is 0, the quadrapole will not depend on the choice of the origin of the system of coordinates.

In the case of point charges where it is easy to describe the degree of dependencies or on the system of coordinates.

:<math>\overrightarrow{\mu} = \sum_i q_i \overrightarrow{r_i}\,\!</math> is the dipole for point charges

But if we have a molecule, the Heisenberg principle does not allow a precise location of the electron. Therefore, wave functions are needed. To get the dipole moment, it is necessary to look at the electronic densities at a given point in space. The expression consists of a wave function at a given point in space r (in other words, wave function at r) and the dipole operator. The dipole operator is the electronic charge of the particles being discussed multiplied by the position of those charges. Since r doesn’t modify Ψ, the Ψ can be arranged so that a psi squared is obtained, which is an electronic density. This makes sense because the wave function squared is an electronic density, and so the expression is equal to an electron density multiplied by a position. Those two expressions are perfectly consistent with one another.

:<math>\overrightarrow{\mu} = \int_{-\infty}^{\infty} \Psi^* e \overrightarrow{r} \Psi d \overrightarrow{r}\,\!</math> is the dipole for a molecule

where
:<math>\Psi^*\,\!</math> is the wave function for position r

:<math>e \overrightarrow{r}\,\!</math> is the dipole operator relating the charge and position of charges.

:<math>\langle \Psi \vert \overrightarrow{r} \vert \Psi \rangle\,\!</math> determines the dipole moment in the state described by the wave function Ψ

:<math>\overrightarrow{\mu}_g = \langle \Psi_g \vert \overrightarrow{r} \vert \Psi_g \rangle\,\!</math> This is just the Dirac expression for this integral expression.

The dipole moment can be examined at the excited state and this is critical for second order materials. To find the dipole moment in the excited state S1 of organic chromaphore, use the wave function of that particular excited state.

=== Transition Dipole ===
The dipole moment should not be confused with the transition dipole in which describes the transition from a given state to another state. Usually it is the transition from ground state to an excited state. In that case, the wave function psi g will be that of the initial state and the wave function psi e will be that of the final state.

:<math>\overrightarrow{\mu}_{eg} = \langle \Psi_e | e \overrightarrow {r} | \Psi)g \rangle\,\!</math> the transition dipole

:<math>\overrightarrow{\mu}_{eg} = \int \Psi^*_e e \overrightarrow{r} \Psi _g d\overrightarrow{r}\,\!</math>

Since r, the operator, multiplies but not modifies the wave function, the r term can be rearranged in this way.

:<math>\overrightarrow{\mu}_{eg} = \int \Psi^*_e (\overrightarrow{r}) \Psi _g (\overrightarrow{r}) e\overrightarrow{r} d\overrightarrow{r}\,\!</math>

These wave functions will be key to the description of polarizabilities in molecular compounds.
In this form the equation shows that in order to have large transition dipoles, it is necessary to have a reasonably good wave function overlap between the wave function of the initial state and the wave function of the final state. At the same time, it is also necessary to have that overlap for large values of r.

This means that if the two charges separate due to the polar attraction of the electric field of the light, it is clear that as the distance between two charges get larger, the dipole will become larger as well. Thus, the r value shows that dipole moment will be larger when the charges separate over a larger distance.

=== Polarizablity to polarization ===

Polarizability refers to a molecule and polarization refers to a material. When a molecule with a very large polarizability or material with a very large polarization is place in an electric field the electrons can move a large distance and delocalize a lot resulting in a separation of charges. Therefore, the larger the distance, the larger the state dipole, and the larger the transition dipole.

We will examine the behavior of individual molecule and then see what happens at the level of the material when many molecules come together.

'''Molecular Level'''

At the microscopic (molecular) level, a molecule that can have a permanent dipole moment , mu0 (permanent dipole) will have an induced dipole moment when it is exposed to electric field (or the electric field of the light). That induced dipole moment is proportional to the electric field an alpha (the polarizability) and the electric field. Thus, we can extend the analogy by acknowledging that the larger the alpha, the larger the polarizability, and the larger the induced dipole moment.

:<math>\overrightarrow{\mu} (induced) = \alpha \cdot \overrightarrow{E}\,\!</math>

where

:<math>\overrightarrow{\mu}\,\!</math> the dipole; a vector

:<math>\overrightarrow{E}\,\!</math> the electric field; a vector

:<math>\alpha\,\!</math> the polarizability is a tensor of rank 2

The electric field, the dipole moment, and induced dipole moment are vectors that have x, y, z components but alpha, the polarizability, is actually a tensor. It is a tensor because if an electric field is applied along x, not only can a dipole moment be generated along x but an induced dipole moment can also be generated along y or z. Therefore the induced dipole moment along axis i will depend on all the components of the field and these polarizability tensor components. So by measuring the induced dipole along mux, the x component of the field can also be observed and an alpha xx tensor component will be obtained.

:<math>\overrightarrow{\mu}_i = \sum_j \alpha_{ij} \overrightarrow{E}_j\,\!</math>

Those polarizability tensor components are a function of the wavelength.

:<math>\alpha_{ij} = f(\omega)\,\!</math>

Usually in the literature one talks about wavelengths and then shows the angular frequency, omega, not lambda. This may be confusing. The reason why the angular frequency is used is simply because the wavelength of the light refers to a very specific oscillation frequency of the electric field of the light.

'''Material Level'''

When looking at the macroscopic scale of the molecular material, one talks about the induced polarization of the material. Just like the microscopic scale, the induced polarization will be directly proportional to the electric field at the macroscopic scale.

:<math>\overrightarrow{P}(induced) = \Epsilon_0 \cdot \Chi_{el} \cdot \overrightarrow{E}\,\!</math>

where

:<math>\Epsilon_0\,\!</math> is the electric permittivity of a vacuum

:<math>\Chi_{el}\,\!</math> is the electric susceptibility of the material; a tensor

The displaced electric (D) field takes into account the external field, like the field due to the light and the polarization.

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} + \overrightarrow{P}\,\!</math>

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} + \Epsilon_0 \cdot \Chi_{el} \cdot \overrightarrow{E}\,\!</math>

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} (1+ \Chi)\,\!</math>

The displaced electric field can be expressed as the electric permittivity of the vacuum times the external field times (1 + χ). (1+χ) represents the electric susceptibility, which is an important characteristic of the material. It is also known as the dielectric constant. Dielectric constant is a function of frequency and wavelength and so it varies as a function of wavelength. Thus, it is constant to some extent. The dielectric constant is actually defined as the ratio of the electric permittivity of the medium itself over that of the vacuum.

:<math>\Epsilon_d = \frac {\Epsilon} {\Epsilon_0}\,\!</math> is the dialectric constant

Therefore, the dielectric constant has no dimension since it is a ratio of those two values. By substituting the ratio in the expression for the displaced electric field, a compact expression can be obtained that expresses the displaced electric field as simply the external field times the electric permittivity of the material.

:<math>\overrightarrow{D} = \Epsilon \overrightarrow{E}\,\!</math>

The permittivity and dielectric constant are a function of frequency (ω) or wavelenght (λ).

:<math>\Epsilon_{ij} (\omega) = 1 + \chi_{ij} (\omega)\,\!</math>

=== Relation between dielectric constant and refractive index ===

It is useful to keep in mind the simple relationship between the refractive index and the dielectric constant. Recall the definition of the speed of light, which is 1 over the square root of the electric permittivity of vacuum times the magnetic permeability of vacuum. When light enters a material (a medium), its speed becomes v, which is 1 over the square root of the product of the electric permittivity of that material or medium times the magnetic permittivity of that material or medium. In this case, mu here is the magnetic permittivity; not the dipole moment.

:<math>c = \frac { 1} {\sqrt{\Epsilon_0 \cdot \mu_o}}\,\!</math>

where
:<math>\Epsilon_0 \,\!</math>is the permittivity of vacuum

:<math>\mu_0 \,\!</math>is the magnetic constant or permeability of vacuum

The ratio of the speed of light in vacuum over that in the in the medium is the index of refraction.

:<math>\frac c v = n =\sqrt {\frac {\Epsilon \mu} {\Epsilon_0 {\mu_0}}}\,\!</math>

In the case of weakly magnetic materials μ = μ0

:<math>n(\omega)\approx \sqrt{\Epsilon_D (\omega)}\,\!</math>

The ratio of the speed of light in vacuum over that in the in the medium is the index of refraction.

:<math>n^2(\omega) \approx \Epsilon_D(\omega)\,\!</math>

For materials in which there is no magnetic activity, the magnetic permeability of the material is basically the magnetic permeability of the vacuum. Then the ratio of μ to μ0 will be close to one and thus the index of refraction will be nearly equal to the square root of the electric permittivity of the medium over that of the vacuum; this is the dielectric constant of the material. Often the interaction of the magnetic field with the light of the material is much smaller than the interaction of the electric field. Therefore, in many instances you can neglect the magnetic interactions and this expression would hold.

Eventually, this comes to an expression where the index of refraction squared corresponds to the dielectric constant. This is the connection between the index of refraction and the dielectric constant. It is important to remember that both the index of refraction and the dielectric constant depend on frequency. Also, often many people in the optics or photonics community casually quote the index of refraction for that material or the dielectric constant you will find that when the two numbers are compared, this expression appears to not hold true. This inconsistency is not due to an issue related to the magnetic material but it is simply because when people casually talk about the dielectric constant or index refraction, they are referring to the constant at very small frequency or in a static context where frequency tends to zero.

When people casually talk about the index refraction of the material, they usually mean optical frequencies. Optical frequencies are frequencies that correspond to the visible. Again, when one casually talks about index of refraction, one would imply optical frequencies where as in dielectric constants, one would imply very low frequencies.

Also, note that n, mu and Epsilon are also complex quantities because they may involve both dispersion effects and attenuation effects. As soon as there is absorption, then that becomes an imaginary component to the expression. Thus, all of those are complex numbers.

Mathematical Expansion of the Dipole Moment

2011-06-28T21:25:40Z

Chrisb:

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Quantum-Mechanical Theory of Molecular Polarizabilities| Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Quantum Mechanical and Perturbation Theory of Polarizability |Return to Quantum Mechanical and Perturbation Theory Menu]]</td>
<td style="text-align: right; width: 33%">[[Perturbation Theory| Next Topic]]</td>
</tr>
</table>
=== Hellman-Feynman Theorem ===

The Hellman-Feynman Theorem, which expresses the dipole moment as minus derivative of the energy of the system with respect to the field. This equation expresses the response of a molecule which is 2nd order in terms of the energy of the molecule and first order in terms of the dipole moment of the molecule (1st order or 2nd order with respect to the field).

:<math>\overrightarrow {\mu} = - \frac {\delta E} {\delta \overrightarrow{F}}\,\!</math>

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} + \frac {1} {2!} \beta \overrightarrow{F}\overrightarrow{F} + \frac {1} {3!} \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\,\!</math>

:<math>\frac {\delta \overrightarrow{\mu}}{\delta \overrightarrow{F}} = \alpha + \beta \overrightarrow{F} + \frac {1} {2!} \gamma \overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

=== Stark Energy Expression ===
Alpha is the linear polarizability or the first order polarizability. It describes how the molecule responds in terms of the modification of its ground state energy or of its dipole moment, in the presence of the field at the limit where the field goes to 0. Thus, α can be cast either as the 1st order derivative of the dipole moment with respect to the field when the field tends to 0 or minus the 2nd order derivative of the ground state energy of the molecule with respect to the field when the field goes to 0.

:<math>\alpha = \left( \frac {\delta\overrightarrow{\mu}}{\delta \overrightarrow{F}}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^2 \overrightarrow{\mu}}{\delta \overrightarrow{F}^2}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

Beta is the 2nd order derivative of the dipole moment with respect to the field when the field goes to 0. Beta will be referred to as the 2nd order polarizability of the molecule. This can also be derived from the stark energy expression which is the 3rd order derivative of the energy with respect to the field when the field tends to 0.

:<math>\beta = \left( \frac {\delta^2 \overrightarrow{\mu}}{\delta \overrightarrow{F}^2}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^3 \overrightarrow{\mu}}{\delta \overrightarrow{F}^3}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

Lastly, the γ term corresponds to the 3rd order derivative of the dipole moment or minus the 4th order derivative of the energy with respect to the field at the limit where the field goes to 0. γ will be referred to as the 3rd order polarizability of the molecule.

:<math>\gamma = \left( \frac {\delta^3 \overrightarrow{\mu}}{\delta \overrightarrow{F}^3}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>
Stark energy describes the evolution of the energy of a system of particles in the presence of an electric field F. In the Stark energy expression, γ corresponds to a 4th order term. However the in common terminology α is referred to as the linear polarizability, β the 2nd order polarizability, and γ the 3rd order polarizability. The Hellman-Feynman Theorem is the origin of these terms. Here it is presented as a Taylor series expansion, sometimes one uses a power series expansion.

:<math>
E_g = E^\circ_g = \overrightarrow{\mu}^\circ \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}\overrightarrow{F} - \frac {1} {3!} \beta \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} - \frac {1} {4!} \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\,\!</math>

:<math>= E^\circ_g - \overrightarrow{\mu}\overrightarrow{F}\,\!</math>

=== Electric dipole approximation ===
The stark energy expression states that the ground state energy of the molecule in the presence of the field is the ground state energy in the absence of the field minus μF. Thus, μF is considered the interaction between the field and the molecule. The electric dipole approximation is that only the electric field component of light influences dipole moment. The magnetic component is ignored. In most instances in the literature, this approximation is assumed but not stated. The expression for the dipole moment is expressed as :

<math>\mu^\circ + \alpha F + \beta F^2 ... \,\!</math>

and so on, without paying any attention to where that expression comes from.

:<math>- \overrightarrow{\mu}\overrightarrow{F}\,\!</math>

in dimensional analysis:
:<math>\mu \equiv\,\!</math> charge * distance
:<math>F \equiv\,\!</math> volt/distance
:<math>\mu F \equiv\,\!</math> charge * volt :<math>\equiv\,\!</math> energy

After the electric dipole, the next two terms are the electric quadrupole and the magnetic dipole. 99% of the time only the electric dipole is considered. This represents the difference in energy between the perturbed state and the unperturbed state of the molecules. Thus, this must have an energy dimension. μ is the dipole moment which is the charge times the distance, and the electric field is volt over distance. When you multiply these terms, the expression has the dimension charge times volt or ev, and ev is an energy unit. This will be used in perturbation theory.

The stark energy expression states that the ground state energy of the molecule in the presence of the field is the ground state energy in the absence of the field minus μF. Thus, μF is considered the interaction between the field and the molecule. The electric dipole approximation is that only the electric field component of light influences dipole moment. The magnetic component is ignored. In most instances in the literature, this approximation is assumed but not stated. The expression for the dipole moment is expressed as

:<math>\mu^\circ + \alpha F + \beta F^2...\,\!</math>

and so on, without paying any attention to where that expression comes from.

:<math>\overrightarrow{\mu} = e \sum(i) \overrightarrow{\pi}_i 9i\,\!</math>

After the electric dipole, the next two terms are the electric quadrupole and the magnetic dipole. 99% of the time only the electric dipole is considered. This represents the difference in energy between the perturbed state and the unperturbed state of the molecules. Thus, this must have an energy dimension. μ is the dipole moment which is the charge times the distance, and the electric field is volt over distance. When you multiply these terms, the expression has the dimension charge times volt or ev, and ev is an energy unit. This will be used in perturbation theory.

Mathematical Expansion of the Dipole Moment

2011-06-28T21:24:38Z

Chrisb: Created page with '<table id="toc" style="width: 100%"> <tr> <td style="text-align: left; width: 33%"> Previous Topic</td> <td style="tex…'

<table id="toc" style="width: 100%">
<tr>
<td style="text-align: left; width: 33%">[[Quantum-Mechanical Theory of Molecular Polarizabilities| Previous Topic]]</td>
<td style="text-align: center; width: 33%">[[Main_Page#Quantum Mechanical and Perturbation Theory of Polarizability |Return to Quantum Mechanical and Perturbation Theory Menu]]</td>

</tr>
</table>
=== Hellman-Feynman Theorem ===

The Hellman-Feynman Theorem, which expresses the dipole moment as minus derivative of the energy of the system with respect to the field. This equation expresses the response of a molecule which is 2nd order in terms of the energy of the molecule and first order in terms of the dipole moment of the molecule (1st order or 2nd order with respect to the field).

:<math>\overrightarrow {\mu} = - \frac {\delta E} {\delta \overrightarrow{F}}\,\!</math>

:<math>\overrightarrow{\mu} = \overrightarrow{\mu^\circ} + \alpha \overrightarrow{F} + \frac {1} {2!} \beta \overrightarrow{F}\overrightarrow{F} + \frac {1} {3!} \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\,\!</math>

:<math>\frac {\delta \overrightarrow{\mu}}{\delta \overrightarrow{F}} = \alpha + \beta \overrightarrow{F} + \frac {1} {2!} \gamma \overrightarrow{F}\overrightarrow{F} + ...\,\!</math>

=== Stark Energy Expression ===
Alpha is the linear polarizability or the first order polarizability. It describes how the molecule responds in terms of the modification of its ground state energy or of its dipole moment, in the presence of the field at the limit where the field goes to 0. Thus, α can be cast either as the 1st order derivative of the dipole moment with respect to the field when the field tends to 0 or minus the 2nd order derivative of the ground state energy of the molecule with respect to the field when the field goes to 0.

:<math>\alpha = \left( \frac {\delta\overrightarrow{\mu}}{\delta \overrightarrow{F}}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^2 \overrightarrow{\mu}}{\delta \overrightarrow{F}^2}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

Beta is the 2nd order derivative of the dipole moment with respect to the field when the field goes to 0. Beta will be referred to as the 2nd order polarizability of the molecule. This can also be derived from the stark energy expression which is the 3rd order derivative of the energy with respect to the field when the field tends to 0.

:<math>\beta = \left( \frac {\delta^2 \overrightarrow{\mu}}{\delta \overrightarrow{F}^2}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^3 \overrightarrow{\mu}}{\delta \overrightarrow{F}^3}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

Lastly, the γ term corresponds to the 3rd order derivative of the dipole moment or minus the 4th order derivative of the energy with respect to the field at the limit where the field goes to 0. γ will be referred to as the 3rd order polarizability of the molecule.

:<math>\gamma = \left( \frac {\delta^3 \overrightarrow{\mu}}{\delta \overrightarrow{F}^3}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>

:<math>= - \left( \frac {\delta^4 \overrightarrow{\mu}}{\delta \overrightarrow{F}^4}\right) \overrightarrow{F} \rightarrow 0 \,\!</math>
Stark energy describes the evolution of the energy of a system of particles in the presence of an electric field F. In the Stark energy expression, γ corresponds to a 4th order term. However the in common terminology α is referred to as the linear polarizability, β the 2nd order polarizability, and γ the 3rd order polarizability. The Hellman-Feynman Theorem is the origin of these terms. Here it is presented as a Taylor series expansion, sometimes one uses a power series expansion.

:<math>
E_g = E^\circ_g = \overrightarrow{\mu}^\circ \overrightarrow{F} - \frac {1} {2!} \alpha \overrightarrow{F}\overrightarrow{F} - \frac {1} {3!} \beta \overrightarrow{F}\overrightarrow{F}\overrightarrow{F} - \frac {1} {4!} \gamma \overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\overrightarrow{F}\,\!</math>

:<math>= E^\circ_g - \overrightarrow{\mu}\overrightarrow{F}\,\!</math>

=== Electric dipole approximation ===
The stark energy expression states that the ground state energy of the molecule in the presence of the field is the ground state energy in the absence of the field minus μF. Thus, μF is considered the interaction between the field and the molecule. The electric dipole approximation is that only the electric field component of light influences dipole moment. The magnetic component is ignored. In most instances in the literature, this approximation is assumed but not stated. The expression for the dipole moment is expressed as :

<math>\mu^\circ + \alpha F + \beta F^2 ... \,\!</math>

and so on, without paying any attention to where that expression comes from.

:<math>- \overrightarrow{\mu}\overrightarrow{F}\,\!</math>

in dimensional analysis:
:<math>\mu \equiv\,\!</math> charge * distance
:<math>F \equiv\,\!</math> volt/distance
:<math>\mu F \equiv\,\!</math> charge * volt :<math>\equiv\,\!</math> energy

After the electric dipole, the next two terms are the electric quadrupole and the magnetic dipole. 99% of the time only the electric dipole is considered. This represents the difference in energy between the perturbed state and the unperturbed state of the molecules. Thus, this must have an energy dimension. μ is the dipole moment which is the charge times the distance, and the electric field is volt over distance. When you multiply these terms, the expression has the dimension charge times volt or ev, and ev is an energy unit. This will be used in perturbation theory.

The stark energy expression states that the ground state energy of the molecule in the presence of the field is the ground state energy in the absence of the field minus μF. Thus, μF is considered the interaction between the field and the molecule. The electric dipole approximation is that only the electric field component of light influences dipole moment. The magnetic component is ignored. In most instances in the literature, this approximation is assumed but not stated. The expression for the dipole moment is expressed as

:<math>\mu^\circ + \alpha F + \beta F^2...\,\!</math>

and so on, without paying any attention to where that expression comes from.

:<math>\overrightarrow{\mu} = e \sum(i) \overrightarrow{\pi}_i 9i\,\!</math>

After the electric dipole, the next two terms are the electric quadrupole and the magnetic dipole. 99% of the time only the electric dipole is considered. This represents the difference in energy between the perturbed state and the unperturbed state of the molecules. Thus, this must have an energy dimension. μ is the dipole moment which is the charge times the distance, and the electric field is volt over distance. When you multiply these terms, the expression has the dimension charge times volt or ev, and ev is an energy unit. This will be used in perturbation theory.

Main Page

2011-06-28T21:23:04Z

Chrisb:

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Quantum-Mechanical Theory of Molecular Polarizabilities

2011-06-20T19:16:39Z

Chrisb: /* Polarizablity to polarization */

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Quantum-Mechanical Theory of Molecular Polarizabilities up to Third Order

The literature refers to first order, second order and third order polarizabilities. You may also see “first order hyperpolarizability “ which is the same thing as second order polarizability. These conventions may be confusing.

Our goal is to be able to relate the chemical structure and the nature of the pi-conjugated backbone, and the nature of donor and acceptors to the to kind of non-linear response that can be measured or calculated.

The expression for dipole moment is simply the sum of all the point charges over all those point charges of the charge itself times the position of that charge.

=== Dipole moment ===

 [[Image:Dipole_example.png|thumb|300px|]]
The drawing of the charges shows a point charge of +1 on the left side and a point charge of -1 on the right side. The origin of the system of coordinates is in the middle of these charges. The point charge of +1 is -5 angstrom away from the origin and the point charge of -1 is +5 Ångström away on the x-axis. The first step to calculate the dipole moment is to multiply the charge +1 by the position -5 Angstrom for the +1 point charge, which will be -5eA (electron times angstrom). Then do the same for the -1 point charge (-1 * 5) which equals -5eA as well. Finally the sum of these values (-5eA + -5eA) -10eA, which corresponds to about -48 debyes, will be the dipole moment. The literature does not use the SI units of the dipole moment because the SI units use coulomb for the charge and meters for the distance which are macroscopic units. Instead, the dipole moment units use electron units and Angstrom (eA), or Debye.

:<math>Cb \cdot \, m \, or Debye \approx 4.8 q(\vert e \vert) * d( A )\,\!</math>

If there is a donor acceptor compound, a full charge transfer can occur. The drawing shows the two point charges that are 10 angstrom apart with the length of the conjugated bridge between them. Then with one full electron charge transfer across 10 angstrom, a dipole moment of 50 debyes will be obtained (this is quite large). If the donor and acceptor are about 10 angstrom apart and the dipole moment is about 10 debyes, the system has a charge transfer of about 0.2 electrons from the donor to the acceptor assuming it a simple dipole.

This example is a simplified model where the origin of the system of coordinates was placed midway between the two charges. However, what would happen if the origin of the coordinates is placed on the +1 point charge? What will be the dipole moment? To answer this question, the +1 point charge will be multiplied by a distance of 0 (due to the origin of the coordinates) and that will not contribute anything to the dipole moment. The -1 point charge is now at a distance of 10 angstroms away and is multiplied by a charge of -1 which is -10 eA or about -50 debyes (-48 precisely).
Consider the case where the molecule is not neutral but instead a cation is formed with a +2 charge on the left.
If the origin of the system of the coordinates is at the +2 point charge, the calculated dipole moment will be -10 eA. But if the origin of system coordinate is placed on the -1 point charge, the dipole moment will be -20 eA. An important lesson out of these examples is that with a charged system, the dipole moment depends the choice of origin of coordinates. The dipole moment is no longer perfectly defined.

In the previous example in which the point origin of coordinates was placed on the +1 point charge, there was a net charge so the monopole is not 0. Therefore, the dipole and all the other poles (quadrapole, octapol, hexapole etc.) depend on the origin of the system of coordinates. However, if the monopole is 0, then the dipole doesn’t depend on the origin of coordinates. But if the dipole is not 0, which means there is a permanent dipole moment in the molecule, then the dipole and the other poles will depend on the origin of coordinates. If the monopole is 0 and the dipole is 0, due to the central symmetric molecule, then the quadrapole will not depend on the choice of the origin of the system of coordinates. It is only when the molecule doesn’t carry a net charge that the dipole is really defined. Otherwise, if the molecule does carry a net charge, the dipole is not defined and it will depend on the origin of the system of coordinates that is chosen. This also creates further complications as it is important to make sure that the point of references is the same for similar systems in order to make comparisons. But for now, when one talks about dipoles, the dipoles will be from neutral molecules so that there will be no complications. In the case of two photon absorption molecules the quadrupole moments can be important. But since these molecules are usually centro symmetric and the dipole is 0, the quadrapole will not depend on the choice of the origin of the system of coordinates.

In the case of point charges where it is easy to describe the degree of dependencies or on the system of coordinates.

:<math>\overrightarrow{\mu} = \sum_i q_i \overrightarrow{r_i}\,\!</math> is the dipole for point charges

But if we have a molecule, the Heisenberg principle does not allow a precise location of the electron. Therefore, wave functions are needed. To get the dipole moment, it is necessary to look at the electronic densities at a given point in space. The expression consists of a wave function at a given point in space r (in other words, wave function at r) and the dipole operator. The dipole operator is the electronic charge of the particles being discussed multiplied by the position of those charges. Since r doesn’t modify Ψ, the Ψ can be arranged so that a psi squared is obtained, which is an electronic density. This makes sense because the wave function squared is an electronic density, and so the expression is equal to an electron density multiplied by a position. Those two expressions are perfectly consistent with one another.

:<math>\overrightarrow{\mu} = \int_{-\infty}^{\infty} \Psi^* e \overrightarrow{r} \Psi d \overrightarrow{r}\,\!</math> is the dipole for a molecule

where
:<math>\Psi^*\,\!</math> is the wave function for position r

:<math>e \overrightarrow{r}\,\!</math> is the dipole operator relating the charge and position of charges.

:<math>\langle \Psi \vert \overrightarrow{r} \vert \Psi \rangle\,\!</math> determines the dipole moment in the state described by the wave function Ψ

:<math>\overrightarrow{\mu}_g = \langle \Psi_g \vert \overrightarrow{r} \vert \Psi_g \rangle\,\!</math> This is just the Dirac expression for this integral expression.

The dipole moment can be examined at the excited state and this is critical for second order materials. To find the dipole moment in the excited state S1 of organic chromaphore, use the wave function of that particular excited state.

=== Transition Dipole ===
The dipole moment should not be confused with the transition dipole in which describes the transition from a given state to another state. Usually it is the transition from ground state to an excited state. In that case, the wave function psi g will be that of the initial state and the wave function psi e will be that of the final state.

:<math>\overrightarrow{\mu}_{eg} = \langle \Psi_e | e \overrightarrow {r} | \Psi)g \rangle\,\!</math> the transition dipole

:<math>\overrightarrow{\mu}_{eg} = \int \Psi^*_e e \overrightarrow{r} \Psi _g d\overrightarrow{r}\,\!</math>

Since r, the operator, multiplies but not modifies the wave function, the r term can be rearranged in this way.

:<math>\overrightarrow{\mu}_{eg} = \int \Psi^*_e (\overrightarrow{r}) \Psi _g (\overrightarrow{r}) e\overrightarrow{r} d\overrightarrow{r}\,\!</math>

These wave functions will be key to the description of polarizabilities in molecular compounds.
In this form the equation shows that in order to have large transition dipoles, it is necessary to have a reasonably good wave function overlap between the wave function of the initial state and the wave function of the final state. At the same time, it is also necessary to have that overlap for large values of r.

This means that if the two charges separate due to the polar attraction of the electric field of the light, it is clear that as the distance between two charges get larger, the dipole will become larger as well. Thus, the r value shows that dipole moment will be larger when the charges separate over a larger distance.

=== Polarizablity to polarization ===

Polarizability refers to a molecule and polarization refers to a material. When a molecule with a very large polarizability or material with a very large polarization is place in an electric field the electrons can move a large distance and delocalize a lot resulting in a separation of charges. Therefore, the larger the distance, the larger the state dipole, and the larger the transition dipole.

We will examine the behavior of individual molecule and then see what happens at the level of the material when many molecules come together.

'''Molecular Level'''

At the microscopic (molecular) level, a molecule that can have a permanent dipole moment , mu0 (permanent dipole) will have an induced dipole moment when it is exposed to electric field (or the electric field of the light). That induced dipole moment is proportional to the electric field an alpha (the polarizability) and the electric field. Thus, we can extend the analogy by acknowledging that the larger the alpha, the larger the polarizability, and the larger the induced dipole moment.

:<math>\overrightarrow{\mu} (induced) = \alpha \cdot \overrightarrow{E}\,\!</math>

where

:<math>\overrightarrow{\mu}\,\!</math> the dipole; a vector

:<math>\overrightarrow{E}\,\!</math> the electric field; a vector

:<math>\alpha\,\!</math> the polarizability is a tensor of rank 2

The electric field, the dipole moment, and induced dipole moment are vectors that have x, y, z components but alpha, the polarizability, is actually a tensor. It is a tensor because if an electric field is applied along x, not only can a dipole moment be generated along x but an induced dipole moment can also be generated along y or z. Therefore the induced dipole moment along axis i will depend on all the components of the field and these polarizability tensor components. So by measuring the induced dipole along mux, the x component of the field can also be observed and an alpha xx tensor component will be obtained.

:<math>\overrightarrow{\mu}_i = \sum_j \alpha_{ij} \overrightarrow{E}_j\,\!</math>

Those polarizability tensor components are a function of the wavelength.

:<math>\alpha_{ij} = f(\omega)\,\!</math>

Usually in the literature one talks about wavelengths and then shows the angular frequency, omega, not lambda. This may be confusing. The reason why the angular frequency is used is simply because the wavelength of the light refers to a very specific oscillation frequency of the electric field of the light.

'''Material Level'''

When looking at the macroscopic scale of the molecular material, one talks about the induced polarization of the material. Just like the microscopic scale, the induced polarization will be directly proportional to the electric field at the macroscopic scale.

:<math>\overrightarrow{P}(induced) = \Epsilon_0 \cdot \Chi_{el} \cdot \overrightarrow{E}\,\!</math>

where

:<math>\Epsilon_0\,\!</math> is the electric permittivity of a vacuum

:<math>\Chi_{el}\,\!</math> is the electric susceptibility of the material; a tensor

The displaced electric (D) field takes into account the external field, like the field due to the light and the polarization.

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} + \overrightarrow{P}\,\!</math>

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} + \Epsilon_0 \cdot \Chi_{el} \cdot \overrightarrow{E}\,\!</math>

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} (1+ \Chi)\,\!</math>

The displaced electric field can be expressed as the electric permittivity of the vacuum times the external field times (1 + χ). (1+χ) represents the electric susceptibility, which is an important characteristic of the material. It is also known as the dielectric constant. Dielectric constant is a function of frequency and wavelength and so it varies as a function of wavelength. Thus, it is constant to some extent. The dielectric constant is actually defined as the ratio of the electric permittivity of the medium itself over that of the vacuum.

:<math>\Epsilon_d = \frac {\Epsilon} {\Epsilon_0}\,\!</math> is the dialectric constant

Therefore, the dielectric constant has no dimension since it is a ratio of those two values. By substituting the ratio in the expression for the displaced electric field, a compact expression can be obtained that expresses the displaced electric field as simply the external field times the electric permittivity of the material.

:<math>\overrightarrow{D} = \Epsilon \overrightarrow{E}\,\!</math>

The permittivity and dielectric constant are a function of frequency (ω) or wavelenght (λ).

:<math>\Epsilon_{ij} (\omega) = 1 + \chi_{ij} (\omega)\,\!</math>

=== Relation between dielectric constant and refractive index ===

It is useful to keep in mind the simple relationship between the refractive index and the dielectric constant. Recall the definition of the speed of light, which is 1 over the square root of the electric permittivity of vacuum times the magnetic permeability of vacuum. When light enters a material (a medium), its speed becomes v, which is 1 over the square root of the product of the electric permittivity of that material or medium times the magnetic permittivity of that material or medium. In this case, mu here is the magnetic permittivity; not the dipole moment.

:<math>c = \frac { 1} {\sqrt{\Epsilon_0 \cdot \mu_o}}\,\!</math>

where
:<math>\Epsilon_0 \,\!</math>is the permittivity of vacuum

:<math>\mu_0 \,\!</math>is the magnetic constant or permeability of vacuum

The ratio of the speed of light in vacuum over that in the in the medium is the index of refraction.

:<math>\frac c v = n =\sqrt {\frac {\Epsilon \mu} {\Epsilon_0 {\mu_0}}}\,\!</math>

In the case of weakly magnetic materials μ = μ0

:<math>n(\omega)\approx \sqrt{\Epsilon_D (\omega)}\,\!</math>

The ratio of the speed of light in vacuum over that in the in the medium is the index of refraction.

:<math>n^2(\omega) \approx \Epsilon_D(\omega)\,\!</math>

For materials in which there is no magnetic activity, the magnetic permeability of the material is basically the magnetic permeability of the vacuum. Then the ratio of μ to μ0 will be close to one and thus the index of refraction will be nearly equal to the square root of the electric permittivity of the medium over that of the vacuum; this is the dielectric constant of the material. Often the interaction of the magnetic field with the light of the material is much smaller than the interaction of the electric field. Therefore, in many instances you can neglect the magnetic interactions and this expression would hold.

Eventually, this comes to an expression where the index of refraction squared corresponds to the dielectric constant. This is the connection between the index of refraction and the dielectric constant. It is important to remember that both the index of refraction and the dielectric constant depend on frequency. Also, often many people in the optics or photonics community casually quote the index of refraction for that material or the dielectric constant you will find that when the two numbers are compared, this expression appears to not hold true. This inconsistency is not due to an issue related to the magnetic material but it is simply because when people casually talk about the dielectric constant or index refraction, they are referring to the constant at very small frequency or in a static context where frequency tends to zero.

When people casually talk about the index refraction of the material, they usually mean optical frequencies. Optical frequencies are frequencies that correspond to the visible. Again, when one casually talks about index of refraction, one would imply optical frequencies where as in dielectric constants, one would imply very low frequencies.

Also, note that n, mu and Epsilon are also complex quantities because they may involve both dispersion effects and attenuation effects. As soon as there is absorption, then that becomes an imaginary component to the expression. Thus, all of those are complex numbers.

Quantum-Mechanical Theory of Molecular Polarizabilities

2011-06-20T19:16:06Z

Chrisb: /* Polarizablity to polarization */

<table id="toc" style="width: 100%">
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<td style="text-align: center; width: 33%">[[Main_Page#Quantum Mechanical and Perturbation Theory of Polarizability |Return to Quantum Mechanical and Perturbation Theory Menu]]</td>
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Quantum-Mechanical Theory of Molecular Polarizabilities up to Third Order

The literature refers to first order, second order and third order polarizabilities. You may also see “first order hyperpolarizability “ which is the same thing as second order polarizability. These conventions may be confusing.

Our goal is to be able to relate the chemical structure and the nature of the pi-conjugated backbone, and the nature of donor and acceptors to the to kind of non-linear response that can be measured or calculated.

The expression for dipole moment is simply the sum of all the point charges over all those point charges of the charge itself times the position of that charge.

=== Dipole moment ===

 [[Image:Dipole_example.png|thumb|300px|]]
The drawing of the charges shows a point charge of +1 on the left side and a point charge of -1 on the right side. The origin of the system of coordinates is in the middle of these charges. The point charge of +1 is -5 angstrom away from the origin and the point charge of -1 is +5 Ångström away on the x-axis. The first step to calculate the dipole moment is to multiply the charge +1 by the position -5 Angstrom for the +1 point charge, which will be -5eA (electron times angstrom). Then do the same for the -1 point charge (-1 * 5) which equals -5eA as well. Finally the sum of these values (-5eA + -5eA) -10eA, which corresponds to about -48 debyes, will be the dipole moment. The literature does not use the SI units of the dipole moment because the SI units use coulomb for the charge and meters for the distance which are macroscopic units. Instead, the dipole moment units use electron units and Angstrom (eA), or Debye.

:<math>Cb \cdot \, m \, or Debye \approx 4.8 q(\vert e \vert) * d( A )\,\!</math>

If there is a donor acceptor compound, a full charge transfer can occur. The drawing shows the two point charges that are 10 angstrom apart with the length of the conjugated bridge between them. Then with one full electron charge transfer across 10 angstrom, a dipole moment of 50 debyes will be obtained (this is quite large). If the donor and acceptor are about 10 angstrom apart and the dipole moment is about 10 debyes, the system has a charge transfer of about 0.2 electrons from the donor to the acceptor assuming it a simple dipole.

This example is a simplified model where the origin of the system of coordinates was placed midway between the two charges. However, what would happen if the origin of the coordinates is placed on the +1 point charge? What will be the dipole moment? To answer this question, the +1 point charge will be multiplied by a distance of 0 (due to the origin of the coordinates) and that will not contribute anything to the dipole moment. The -1 point charge is now at a distance of 10 angstroms away and is multiplied by a charge of -1 which is -10 eA or about -50 debyes (-48 precisely).
Consider the case where the molecule is not neutral but instead a cation is formed with a +2 charge on the left.
If the origin of the system of the coordinates is at the +2 point charge, the calculated dipole moment will be -10 eA. But if the origin of system coordinate is placed on the -1 point charge, the dipole moment will be -20 eA. An important lesson out of these examples is that with a charged system, the dipole moment depends the choice of origin of coordinates. The dipole moment is no longer perfectly defined.

In the previous example in which the point origin of coordinates was placed on the +1 point charge, there was a net charge so the monopole is not 0. Therefore, the dipole and all the other poles (quadrapole, octapol, hexapole etc.) depend on the origin of the system of coordinates. However, if the monopole is 0, then the dipole doesn’t depend on the origin of coordinates. But if the dipole is not 0, which means there is a permanent dipole moment in the molecule, then the dipole and the other poles will depend on the origin of coordinates. If the monopole is 0 and the dipole is 0, due to the central symmetric molecule, then the quadrapole will not depend on the choice of the origin of the system of coordinates. It is only when the molecule doesn’t carry a net charge that the dipole is really defined. Otherwise, if the molecule does carry a net charge, the dipole is not defined and it will depend on the origin of the system of coordinates that is chosen. This also creates further complications as it is important to make sure that the point of references is the same for similar systems in order to make comparisons. But for now, when one talks about dipoles, the dipoles will be from neutral molecules so that there will be no complications. In the case of two photon absorption molecules the quadrupole moments can be important. But since these molecules are usually centro symmetric and the dipole is 0, the quadrapole will not depend on the choice of the origin of the system of coordinates.

In the case of point charges where it is easy to describe the degree of dependencies or on the system of coordinates.

:<math>\overrightarrow{\mu} = \sum_i q_i \overrightarrow{r_i}\,\!</math> is the dipole for point charges

But if we have a molecule, the Heisenberg principle does not allow a precise location of the electron. Therefore, wave functions are needed. To get the dipole moment, it is necessary to look at the electronic densities at a given point in space. The expression consists of a wave function at a given point in space r (in other words, wave function at r) and the dipole operator. The dipole operator is the electronic charge of the particles being discussed multiplied by the position of those charges. Since r doesn’t modify Ψ, the Ψ can be arranged so that a psi squared is obtained, which is an electronic density. This makes sense because the wave function squared is an electronic density, and so the expression is equal to an electron density multiplied by a position. Those two expressions are perfectly consistent with one another.

:<math>\overrightarrow{\mu} = \int_{-\infty}^{\infty} \Psi^* e \overrightarrow{r} \Psi d \overrightarrow{r}\,\!</math> is the dipole for a molecule

where
:<math>\Psi^*\,\!</math> is the wave function for position r

:<math>e \overrightarrow{r}\,\!</math> is the dipole operator relating the charge and position of charges.

:<math>\langle \Psi \vert \overrightarrow{r} \vert \Psi \rangle\,\!</math> determines the dipole moment in the state described by the wave function Ψ

:<math>\overrightarrow{\mu}_g = \langle \Psi_g \vert \overrightarrow{r} \vert \Psi_g \rangle\,\!</math> This is just the Dirac expression for this integral expression.

The dipole moment can be examined at the excited state and this is critical for second order materials. To find the dipole moment in the excited state S1 of organic chromaphore, use the wave function of that particular excited state.

=== Transition Dipole ===
The dipole moment should not be confused with the transition dipole in which describes the transition from a given state to another state. Usually it is the transition from ground state to an excited state. In that case, the wave function psi g will be that of the initial state and the wave function psi e will be that of the final state.

:<math>\overrightarrow{\mu}_{eg} = \langle \Psi_e | e \overrightarrow {r} | \Psi)g \rangle\,\!</math> the transition dipole

:<math>\overrightarrow{\mu}_{eg} = \int \Psi^*_e e \overrightarrow{r} \Psi _g d\overrightarrow{r}\,\!</math>

Since r, the operator, multiplies but not modifies the wave function, the r term can be rearranged in this way.

:<math>\overrightarrow{\mu}_{eg} = \int \Psi^*_e (\overrightarrow{r}) \Psi _g (\overrightarrow{r}) e\overrightarrow{r} d\overrightarrow{r}\,\!</math>

These wave functions will be key to the description of polarizabilities in molecular compounds.
In this form the equation shows that in order to have large transition dipoles, it is necessary to have a reasonably good wave function overlap between the wave function of the initial state and the wave function of the final state. At the same time, it is also necessary to have that overlap for large values of r.

This means that if the two charges separate due to the polar attraction of the electric field of the light, it is clear that as the distance between two charges get larger, the dipole will become larger as well. Thus, the r value shows that dipole moment will be larger when the charges separate over a larger distance.

=== Polarizablity to polarization ===

Polarizability refers to a molecule and polarization refers to a material. When a molecule with a very large polarizability or material with a very large polarization is place in an electric field the electrons can move a large distance and delocalize a lot resulting in a separation of charges. Therefore, the larger the distance, the larger the state dipole, and the larger the transition dipole.

We will examine the behavior of individual molecule and then see what happens at the level of the material when many molecules come together.

'''Molecular Level'''

At the microscopic (molecular) level, a molecule that can have a permanent dipole moment , mu0 (permanent dipole) will have an induced dipole moment when it is exposed to electric field (or the electric field of the light). That induced dipole moment is proportional to the electric field an alpha (the polarizability) and the electric field. Thus, we can extend the analogy by acknowledging that the larger the alpha, the larger the polarizability, and the larger the induced dipole moment.

:<math>\overrightarrow{\mu} (induced) = \alpha \cdot \overrightarrow{E}\,\!</math>

where

:<math>\overrightarrow{\mu}\,\!</math> the dipole; a vector

:<math>\overrightarrow{E}\,\!</math> the electric field; a vector

:<math>\alpha\,\!</math> the polarizability is a tensor of rank 2

The electric field, the dipole moment, and induced dipole moment are vectors that have x, y, z components but alpha, the polarizability, is actually a tensor. It is a tensor because if an electric field is applied along x, not only can a dipole moment be generated along x but an induced dipole moment can also be generated along y or z. Therefore the induced dipole moment along axis i will depend on all the components of the field and these polarizability tensor components. So by measuring the induced dipole along mux, the x component of the field can also be observed and an alpha xx tensor component will be obtained.

:<math>\overrightarrow{\mu}_i = \sum_j \alpha_{ij} \overrightarrow{E}_j\,\!</math>

Those polarizability tensor components are a function of the wavelength.

:<math>\alpha_{ij} = f(\omega)\,\!</math>

Usually in the literature one talks about wavelengths and then shows the angular frequency, omega, not lambda. This may be confusing. The reason why the angular frequency is used is simply because the wavelength of the light refers to a very specific oscillation frequency of the electric field of the light.

'''Material Level'''

When looking at the macroscopic scale of the molecular material, one talks about the induced polarization of the material. Just like the microscopic scale, the induced polarization will be directly proportional to the electric field at the macroscopic scale.

:<math>\overrightarrow{P}(induced) = \Epsilon_0 \cdot \Chi_{el} \cdot \overrightarrow{E}\,\!</math>

where

:<math>\epsilon_0\,\!</math> is the electric permittivity of a vacuum

:<math>\Chi_{el}\,\!</math> is the electric susceptibility of the material; a tensor

The displaced electric (D) field takes into account the external field, like the field due to the light and the polarization.

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} + \overrightarrow{P}\,\!</math>

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} + \Epsilon_0 \cdot \Chi_{el} \cdot \overrightarrow{E}\,\!</math>

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} (1+ \Chi)\,\!</math>

The displaced electric field can be expressed as the electric permittivity of the vacuum times the external field times (1 + χ). (1+χ) represents the electric susceptibility, which is an important characteristic of the material. It is also known as the dielectric constant. Dielectric constant is a function of frequency and wavelength and so it varies as a function of wavelength. Thus, it is constant to some extent. The dielectric constant is actually defined as the ratio of the electric permittivity of the medium itself over that of the vacuum.

:<math>\Epsilon_d = \frac {\Epsilon} {\Epsilon_0}\,\!</math> is the dialectric constant

Therefore, the dielectric constant has no dimension since it is a ratio of those two values. By substituting the ratio in the expression for the displaced electric field, a compact expression can be obtained that expresses the displaced electric field as simply the external field times the electric permittivity of the material.

:<math>\overrightarrow{D} = \Epsilon \overrightarrow{E}\,\!</math>

The permittivity and dielectric constant are a function of frequency (ω) or wavelenght (λ).

:<math>\Epsilon_{ij} (\omega) = 1 + \chi_{ij} (\omega)\,\!</math>

=== Relation between dielectric constant and refractive index ===

It is useful to keep in mind the simple relationship between the refractive index and the dielectric constant. Recall the definition of the speed of light, which is 1 over the square root of the electric permittivity of vacuum times the magnetic permeability of vacuum. When light enters a material (a medium), its speed becomes v, which is 1 over the square root of the product of the electric permittivity of that material or medium times the magnetic permittivity of that material or medium. In this case, mu here is the magnetic permittivity; not the dipole moment.

:<math>c = \frac { 1} {\sqrt{\Epsilon_0 \cdot \mu_o}}\,\!</math>

where
:<math>\Epsilon_0 \,\!</math>is the permittivity of vacuum

:<math>\mu_0 \,\!</math>is the magnetic constant or permeability of vacuum

The ratio of the speed of light in vacuum over that in the in the medium is the index of refraction.

:<math>\frac c v = n =\sqrt {\frac {\Epsilon \mu} {\Epsilon_0 {\mu_0}}}\,\!</math>

In the case of weakly magnetic materials μ = μ0

:<math>n(\omega)\approx \sqrt{\Epsilon_D (\omega)}\,\!</math>

The ratio of the speed of light in vacuum over that in the in the medium is the index of refraction.

:<math>n^2(\omega) \approx \Epsilon_D(\omega)\,\!</math>

For materials in which there is no magnetic activity, the magnetic permeability of the material is basically the magnetic permeability of the vacuum. Then the ratio of μ to μ0 will be close to one and thus the index of refraction will be nearly equal to the square root of the electric permittivity of the medium over that of the vacuum; this is the dielectric constant of the material. Often the interaction of the magnetic field with the light of the material is much smaller than the interaction of the electric field. Therefore, in many instances you can neglect the magnetic interactions and this expression would hold.

Eventually, this comes to an expression where the index of refraction squared corresponds to the dielectric constant. This is the connection between the index of refraction and the dielectric constant. It is important to remember that both the index of refraction and the dielectric constant depend on frequency. Also, often many people in the optics or photonics community casually quote the index of refraction for that material or the dielectric constant you will find that when the two numbers are compared, this expression appears to not hold true. This inconsistency is not due to an issue related to the magnetic material but it is simply because when people casually talk about the dielectric constant or index refraction, they are referring to the constant at very small frequency or in a static context where frequency tends to zero.

When people casually talk about the index refraction of the material, they usually mean optical frequencies. Optical frequencies are frequencies that correspond to the visible. Again, when one casually talks about index of refraction, one would imply optical frequencies where as in dielectric constants, one would imply very low frequencies.

Also, note that n, mu and Epsilon are also complex quantities because they may involve both dispersion effects and attenuation effects. As soon as there is absorption, then that becomes an imaginary component to the expression. Thus, all of those are complex numbers.

Quantum-Mechanical Theory of Molecular Polarizabilities

2011-06-20T19:14:51Z

Chrisb: /* Polarizablity to polarization */

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Quantum-Mechanical Theory of Molecular Polarizabilities up to Third Order

The literature refers to first order, second order and third order polarizabilities. You may also see “first order hyperpolarizability “ which is the same thing as second order polarizability. These conventions may be confusing.

Our goal is to be able to relate the chemical structure and the nature of the pi-conjugated backbone, and the nature of donor and acceptors to the to kind of non-linear response that can be measured or calculated.

The expression for dipole moment is simply the sum of all the point charges over all those point charges of the charge itself times the position of that charge.

=== Dipole moment ===

 [[Image:Dipole_example.png|thumb|300px|]]
The drawing of the charges shows a point charge of +1 on the left side and a point charge of -1 on the right side. The origin of the system of coordinates is in the middle of these charges. The point charge of +1 is -5 angstrom away from the origin and the point charge of -1 is +5 Ångström away on the x-axis. The first step to calculate the dipole moment is to multiply the charge +1 by the position -5 Angstrom for the +1 point charge, which will be -5eA (electron times angstrom). Then do the same for the -1 point charge (-1 * 5) which equals -5eA as well. Finally the sum of these values (-5eA + -5eA) -10eA, which corresponds to about -48 debyes, will be the dipole moment. The literature does not use the SI units of the dipole moment because the SI units use coulomb for the charge and meters for the distance which are macroscopic units. Instead, the dipole moment units use electron units and Angstrom (eA), or Debye.

:<math>Cb \cdot \, m \, or Debye \approx 4.8 q(\vert e \vert) * d( A )\,\!</math>

If there is a donor acceptor compound, a full charge transfer can occur. The drawing shows the two point charges that are 10 angstrom apart with the length of the conjugated bridge between them. Then with one full electron charge transfer across 10 angstrom, a dipole moment of 50 debyes will be obtained (this is quite large). If the donor and acceptor are about 10 angstrom apart and the dipole moment is about 10 debyes, the system has a charge transfer of about 0.2 electrons from the donor to the acceptor assuming it a simple dipole.

This example is a simplified model where the origin of the system of coordinates was placed midway between the two charges. However, what would happen if the origin of the coordinates is placed on the +1 point charge? What will be the dipole moment? To answer this question, the +1 point charge will be multiplied by a distance of 0 (due to the origin of the coordinates) and that will not contribute anything to the dipole moment. The -1 point charge is now at a distance of 10 angstroms away and is multiplied by a charge of -1 which is -10 eA or about -50 debyes (-48 precisely).
Consider the case where the molecule is not neutral but instead a cation is formed with a +2 charge on the left.
If the origin of the system of the coordinates is at the +2 point charge, the calculated dipole moment will be -10 eA. But if the origin of system coordinate is placed on the -1 point charge, the dipole moment will be -20 eA. An important lesson out of these examples is that with a charged system, the dipole moment depends the choice of origin of coordinates. The dipole moment is no longer perfectly defined.

In the previous example in which the point origin of coordinates was placed on the +1 point charge, there was a net charge so the monopole is not 0. Therefore, the dipole and all the other poles (quadrapole, octapol, hexapole etc.) depend on the origin of the system of coordinates. However, if the monopole is 0, then the dipole doesn’t depend on the origin of coordinates. But if the dipole is not 0, which means there is a permanent dipole moment in the molecule, then the dipole and the other poles will depend on the origin of coordinates. If the monopole is 0 and the dipole is 0, due to the central symmetric molecule, then the quadrapole will not depend on the choice of the origin of the system of coordinates. It is only when the molecule doesn’t carry a net charge that the dipole is really defined. Otherwise, if the molecule does carry a net charge, the dipole is not defined and it will depend on the origin of the system of coordinates that is chosen. This also creates further complications as it is important to make sure that the point of references is the same for similar systems in order to make comparisons. But for now, when one talks about dipoles, the dipoles will be from neutral molecules so that there will be no complications. In the case of two photon absorption molecules the quadrupole moments can be important. But since these molecules are usually centro symmetric and the dipole is 0, the quadrapole will not depend on the choice of the origin of the system of coordinates.

In the case of point charges where it is easy to describe the degree of dependencies or on the system of coordinates.

:<math>\overrightarrow{\mu} = \sum_i q_i \overrightarrow{r_i}\,\!</math> is the dipole for point charges

But if we have a molecule, the Heisenberg principle does not allow a precise location of the electron. Therefore, wave functions are needed. To get the dipole moment, it is necessary to look at the electronic densities at a given point in space. The expression consists of a wave function at a given point in space r (in other words, wave function at r) and the dipole operator. The dipole operator is the electronic charge of the particles being discussed multiplied by the position of those charges. Since r doesn’t modify Ψ, the Ψ can be arranged so that a psi squared is obtained, which is an electronic density. This makes sense because the wave function squared is an electronic density, and so the expression is equal to an electron density multiplied by a position. Those two expressions are perfectly consistent with one another.

:<math>\overrightarrow{\mu} = \int_{-\infty}^{\infty} \Psi^* e \overrightarrow{r} \Psi d \overrightarrow{r}\,\!</math> is the dipole for a molecule

where
:<math>\Psi^*\,\!</math> is the wave function for position r

:<math>e \overrightarrow{r}\,\!</math> is the dipole operator relating the charge and position of charges.

:<math>\langle \Psi \vert \overrightarrow{r} \vert \Psi \rangle\,\!</math> determines the dipole moment in the state described by the wave function Ψ

:<math>\overrightarrow{\mu}_g = \langle \Psi_g \vert \overrightarrow{r} \vert \Psi_g \rangle\,\!</math> This is just the Dirac expression for this integral expression.

The dipole moment can be examined at the excited state and this is critical for second order materials. To find the dipole moment in the excited state S1 of organic chromaphore, use the wave function of that particular excited state.

=== Transition Dipole ===
The dipole moment should not be confused with the transition dipole in which describes the transition from a given state to another state. Usually it is the transition from ground state to an excited state. In that case, the wave function psi g will be that of the initial state and the wave function psi e will be that of the final state.

:<math>\overrightarrow{\mu}_{eg} = \langle \Psi_e | e \overrightarrow {r} | \Psi)g \rangle\,\!</math> the transition dipole

:<math>\overrightarrow{\mu}_{eg} = \int \Psi^*_e e \overrightarrow{r} \Psi _g d\overrightarrow{r}\,\!</math>

Since r, the operator, multiplies but not modifies the wave function, the r term can be rearranged in this way.

:<math>\overrightarrow{\mu}_{eg} = \int \Psi^*_e (\overrightarrow{r}) \Psi _g (\overrightarrow{r}) e\overrightarrow{r} d\overrightarrow{r}\,\!</math>

These wave functions will be key to the description of polarizabilities in molecular compounds.
In this form the equation shows that in order to have large transition dipoles, it is necessary to have a reasonably good wave function overlap between the wave function of the initial state and the wave function of the final state. At the same time, it is also necessary to have that overlap for large values of r.

This means that if the two charges separate due to the polar attraction of the electric field of the light, it is clear that as the distance between two charges get larger, the dipole will become larger as well. Thus, the r value shows that dipole moment will be larger when the charges separate over a larger distance.

=== Polarizablity to polarization ===

Polarizability refers to a molecule and polarization refers to a material. When a molecule with a very large polarizability or material with a very large polarization is place in an electric field the electrons can move a large distance and delocalize a lot resulting in a separation of charges. Therefore, the larger the distance, the larger the state dipole, and the larger the transition dipole.

We will examine the behavior of individual molecule and then see what happens at the level of the material when many molecules come together.

'''Molecular Level'''

At the microscopic (molecular) level, a molecule that can have a permanent dipole moment , mu0 (permanent dipole) will have an induced dipole moment when it is exposed to electric field (or the electric field of the light). That induced dipole moment is proportional to the electric field an alpha (the polarizability) and the electric field. Thus, we can extend the analogy by acknowledging that the larger the alpha, the larger the polarizability, and the larger the induced dipole moment.

:<math>\overrightarrow{\mu} (induced) = \alpha \cdot \overrightarrow{E}\,\!</math>

where

:<math>\overrightarrow{\mu}\,\!</math> the dipole; a vector

:<math>\overrightarrow{E}\,\!</math> the electric field; a vector

:<math>\alpha\,\!</math> the polarizability is a tensor of rank 2

The electric field, the dipole moment, and induced dipole moment are vectors that have x, y, z components but alpha, the polarizability, is actually a tensor. It is a tensor because if an electric field is applied along x, not only can a dipole moment be generated along x but an induced dipole moment can also be generated along y or z. Therefore the induced dipole moment along axis i will depend on all the components of the field and these polarizability tensor components. So by measuring the induced dipole along mux, the x component of the field can also be observed and an alpha xx tensor component will be obtained.

:<math>\overrightarrow{\mu}_i = \sum_j \alpha_{ij} \overrightarrow{E}_j\,\!</math>

Those polarizability tensor components are a function of the wavelength.

:<math>\alpha_{ij} = f(\omega)\,\!</math>

Usually in the literature one talks about wavelengths and then shows the angular frequency, omega, not lambda. This may be confusing. The reason why the angular frequency is used is simply because the wavelength of the light refers to a very specific oscillation frequency of the electric field of the light.

'''Material Level'''

When looking at the macroscopic scale of the molecular material, one talks about the induced polarization of the material. Just like the microscopic scale, the induced polarization will be directly proportional to the electric field at the macroscopic scale.

:<math>\overrightarrow{P}(induced) = \Epsilon_0 \cdot \Chi_{el} \cdot \overrightarrow{E}\,\!</math>

where

:<math>\Epsilon_0\,\!</math> is the electric permittivity of a vacuum

:<math>\Chi_{el}\,\!</math> is the electric susceptibility of the material; a tensor

The displaced electric (D) field takes into account the external field, like the field due to the light and the polarization.

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} + \overrightarrow{P}\,\!</math>

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} + \Epsilon_0 \cdot \Chi_{el} \cdot \overrightarrow{E}\,\!</math>

:<math>\overrightarrow{D} = \Epsilon_0 \overrightarrow{E} (1+ \Chi)\,\!</math>

The displaced electric field can be expressed as the electric permittivity of the vacuum times the external field times (1 + χ). (1+χ) represents the electric susceptibility, which is an important characteristic of the material. It is also known as the dielectric constant. Dielectric constant is a function of frequency and wavelength and so it varies as a function of wavelength. Thus, it is constant to some extent. The dielectric constant is actually defined as the ratio of the electric permittivity of the medium itself over that of the vacuum.

:<math>\Epsilon_d = \frac {\Epsilon} {\Epsilon_0}\,\!</math> is the dialectric constant

Therefore, the dielectric constant has no dimension since it is a ratio of those two values. By substituting the ratio in the expression for the displaced electric field, a compact expression can be obtained that expresses the displaced electric field as simply the external field times the electric permittivity of the material.

:<math>\overrightarrow{D} = \Epsilon \overrightarrow{E}\,\!</math>

The permittivity and dielectric constant are a function of frequency (ω) or wavelenght (λ).

:<math>\Epsilon_{ij} (\omega) = 1 + \chi_{ij} (\omega)\,\!</math>

=== Relation between dielectric constant and refractive index ===

It is useful to keep in mind the simple relationship between the refractive index and the dielectric constant. Recall the definition of the speed of light, which is 1 over the square root of the electric permittivity of vacuum times the magnetic permeability of vacuum. When light enters a material (a medium), its speed becomes v, which is 1 over the square root of the product of the electric permittivity of that material or medium times the magnetic permittivity of that material or medium. In this case, mu here is the magnetic permittivity; not the dipole moment.

:<math>c = \frac { 1} {\sqrt{\Epsilon_0 \cdot \mu_o}}\,\!</math>

where
:<math>\Epsilon_0 \,\!</math>is the permittivity of vacuum

:<math>\mu_0 \,\!</math>is the magnetic constant or permeability of vacuum

The ratio of the speed of light in vacuum over that in the in the medium is the index of refraction.

:<math>\frac c v = n =\sqrt {\frac {\Epsilon \mu} {\Epsilon_0 {\mu_0}}}\,\!</math>

In the case of weakly magnetic materials μ = μ0

:<math>n(\omega)\approx \sqrt{\Epsilon_D (\omega)}\,\!</math>

The ratio of the speed of light in vacuum over that in the in the medium is the index of refraction.

:<math>n^2(\omega) \approx \Epsilon_D(\omega)\,\!</math>

For materials in which there is no magnetic activity, the magnetic permeability of the material is basically the magnetic permeability of the vacuum. Then the ratio of μ to μ0 will be close to one and thus the index of refraction will be nearly equal to the square root of the electric permittivity of the medium over that of the vacuum; this is the dielectric constant of the material. Often the interaction of the magnetic field with the light of the material is much smaller than the interaction of the electric field. Therefore, in many instances you can neglect the magnetic interactions and this expression would hold.

Eventually, this comes to an expression where the index of refraction squared corresponds to the dielectric constant. This is the connection between the index of refraction and the dielectric constant. It is important to remember that both the index of refraction and the dielectric constant depend on frequency. Also, often many people in the optics or photonics community casually quote the index of refraction for that material or the dielectric constant you will find that when the two numbers are compared, this expression appears to not hold true. This inconsistency is not due to an issue related to the magnetic material but it is simply because when people casually talk about the dielectric constant or index refraction, they are referring to the constant at very small frequency or in a static context where frequency tends to zero.

When people casually talk about the index refraction of the material, they usually mean optical frequencies. Optical frequencies are frequencies that correspond to the visible. Again, when one casually talks about index of refraction, one would imply optical frequencies where as in dielectric constants, one would imply very low frequencies.

Also, note that n, mu and Epsilon are also complex quantities because they may involve both dispersion effects and attenuation effects. As soon as there is absorption, then that becomes an imaginary component to the expression. Thus, all of those are complex numbers.