Transition Dipole Moment - Revision history

Chrisb: /* Pseudo-physical description of transition dipole moment */

2011-07-18T22:52:13Z

Pseudo-physical description of transition dipole moment

← Older revision		Revision as of 15:52, 18 July 2011
Line 160:		Line 160:
	Now if the wave functions are considered to be linear combinations of atomic orbitals then:		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>		:<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 ==		== Summary ==

Chrisb: /* Transition dipole moment */

2011-07-18T22:22:03Z

Transition dipole moment

← Older revision		Revision as of 15:22, 18 July 2011
Line 112:		Line 112:

	:<math>		:<math>
	\mu^{2}_{g,e} \propto \int ~~\vert~~ \Psi^{*}_{g} (R) \Psi_e \partial \tau \vert ^2\,\!</math>		\mu^{2}_{g,e} \propto \vert \int \Psi^{*}_{g} (R) \Psi_e \partial \tau \vert ^2\,\!</math>

Cmditradmin: /* Transition Moment Operator */

2010-06-23T16:52:31Z

Transition Moment Operator

← Older revision		Revision as of 09:52, 23 June 2010
Line 136:		Line 136:

	=== Transition Moment Operator ===		=== Transition Moment Operator ===
	[[Image:Ethylene_mo_transitiondipole.png\|thumb\|~~300px~~\|The ground- and excited-state wave function must not have the same symmetry if the transition dipole moment is to be non-zero.]]		[[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.		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.

Cmditradmin at 20:04, 29 December 2009

2009-12-29T20:04:40Z

← Older revision		Revision as of 13:04, 29 December 2009
Line 174:		Line 174:

	<br clear='all'>		<br clear='all'>
			[[category:absorption]]
	<table id="toc" style="width: 100%">		<table id="toc" style="width: 100%">
	<tr>		<tr>

Cmditradmin at 21:00, 7 December 2009

2009-12-07T21:00:41Z

Cmditradmin: /* Quantum Description of Dipole Moment */

2009-09-01T21:10:52Z

Quantum Description of Dipole Moment

← Older revision		Revision as of 14:10, 1 September 2009
Line 87:		Line 87:
	=== Quantum Description of Dipole Moment ===		=== 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.		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 wavefunction 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>		:<math>\mu = \int \Psi^*(\sum_n e_n r_n) \Psi \partial \tau\,\!</math>

Cmditradmin: /* Oscillator Strength and Molecular Parameters */

2009-09-01T21:10:20Z

Oscillator Strength and Molecular Parameters

← Older revision		Revision as of 14:10, 1 September 2009
Line 35:		Line 35:
	Try matching elements with the [http://concave.stc.arizona.edu/thepoint/Interactive/spectrasample.html Emission spectra simulator]		Try matching elements with the [http://concave.stc.arizona.edu/thepoint/Interactive/spectrasample.html Emission spectra simulator]

	== Oscillator Strength and Molecular Parameters ==		=== 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.		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.

Cmditradmin at 21:07, 1 September 2009

2009-09-01T21:07:45Z

← Older revision		Revision as of 14:07, 1 September 2009
Line 7:		Line 7:
	</table>		</table>

			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 ==		== Spectroscopy ==

Cmditradmin: /* Spectroscopy */

2009-09-01T21:04:33Z

Spectroscopy

← Older revision		Revision as of 14:04, 1 September 2009
Line 25:		Line 25:

	:<math>\varepsilon\,\!</math> is the molar absorptivity or extinction coefficient with units of l-mol<sup>-1</sup>cm<sup>-1</sup>.		:<math>\varepsilon\,\!</math> is the molar absorptivity or extinction coefficient with units of l-mol<sup>-1</sup>cm<sup>-1</sup>.


			=== Extinction Coefficient ===

	The extinction coefficient characterizes the ability of a molecule to absorb light at a given wavelength		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 ~~your~~ 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<sup>-1</sup> cm<sup>-1</sup> 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.		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<sup>-1</sup> cm<sup>-1</sup> 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]		Try matching elements with the [http://concave.stc.arizona.edu/thepoint/Interactive/spectrasample.html Emission spectra simulator]

Cmditradmin: /* Spectroscopy */

2009-09-01T16:15:16Z

Spectroscopy

← Older revision		Revision as of 09:15, 1 September 2009
Line 29:		Line 29:

	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 your 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<sup>-1</sup> cm<sup>-1</sup> 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.		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 your 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<sup>-1</sup> cm<sup>-1</sup> 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 ==		== Oscillator Strength and Molecular Parameters ==

← Older revision		Revision as of 14:00, 7 December 2009
Line 46:		Line 46:
	:<math>f \approx 4.32 \times 10^9 \varepsilon_{max} \Delta v_{1/2}\,\!</math>		:<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 ~~wavefunction~~ of the molecules.		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 ==		==Transition dipole moment ==
Line 87:		Line 87:
	=== Quantum Description of Dipole Moment ===		=== 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 ~~wavefunction~~ 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.		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>		:<math>\mu = \int \Psi^*(\sum_n e_n r_n) \Psi \partial \tau\,\!</math>

	Where		Where
	:<math>\Psi^*\,\!</math> is the complex conjugate of the ~~wavefunction~~		:<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>r_n\,\!</math> is the position of the particle with respect to the coordinate system
Line 115:		Line 115:


	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 ~~wavefunction~~ of the molecule back to the oscillator strength and the extinction coefficient.		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.		That integral relates to the strength of the absorption band.

Line 136:		Line 136:

	=== Transition Moment Operator ===		=== Transition Moment Operator ===
	[[Image:Ethylene_mo_transitiondipole.png\|thumb\|300px\|The ground- and excited-state ~~wavefunction~~ must not have the same symmetry if the transition dipole moment is to be non-zero.]]		[[Image:Ethylene_mo_transitiondipole.png\|thumb\|300px\|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 ~~wavefunctions~~ are even or odd, the ground- and excited-state ~~wavefunction~~ 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 ~~wavefunction~~ for a particle in a box upon excitation from the ground state (with no nodes) to the first excited state with one node.		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.
	<br clear='all'>		<br clear='all'>

Line 146:		Line 146:
	[[Image:Wavefunction_boxed.png\|thumb\|300px\|Wavefunction during transition]]		[[Image:Wavefunction_boxed.png\|thumb\|300px\|Wavefunction during transition]]
	Initially the function is symmetric with respect to the axis of the one dimensional box.		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 ~~wavefunction~~ is evolving from the initial to final state.		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 ~~wavefunction~~:		Now consider that the electron density during process is the square of the wave function:


Line 158:		Line 158:
	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.		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 ~~wavefunctions~~ are considered to be linear combinations of atomic orbitals then:		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>		:<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>
Line 165:		Line 165:


	*There needs to be significant spatial overlap throughout the molecule between the ground- and excited-state ~~wavefunction~~, if the transition moment that couples the two states is to be large.		*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.		*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.