Difference between revisions of "Model Calculations of Electronic Coupling"
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splitting HOMO ≡ 2 tHOMO | splitting HOMO ≡ 2 tHOMO | ||
in a 1D stack: WHOMO = 4 tHOMO = valence (hole) bandwidth | |||
splitting LUMO ≡ 2 tLUMO | |||
in a 1D stack: WLUMO = 4 tLUMO = conduction (electron) bandwidth | |||
Revision as of 14:45, 18 June 2009
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Model calculations of electronic coupling
There have not been that many measurements of pure perfect single crystal. Researchers at Stuttgart have characterized oligo- acene single crystals. This clearly shows band regime at low temperature with mobility decreasing with increasing temperature, then eventually a cross over localization and transition to a hopping regime, followed by a increasing mobility.
As you bring two molecules together there is an interaction resulting in the splitting of the HOMO an LUMO levels.
splitting HOMO ≡ 2 tHOMO in a 1D stack: WHOMO = 4 tHOMO = valence (hole) bandwidth
splitting LUMO ≡ 2 tLUMO in a 1D stack: WLUMO = 4 tLUMO = conduction (electron) bandwidth
Influence of intermolecular distance
As the molecules come closer together their wavefunction overlap grows exponentially.
The exponential increase in energy with decreasing distance between the molecules is a manifestation that the splitting is caused by a wavefunction overlap. In the typical solid state intermolecular distances of 3.4 - 3.8 Å given a perfectly symmetrical, cofacial arrangement, there is splitting on the order of several tenths of an eV meaning a bandgap on the order of 1 eV. This consistent with formation of a band regime.
When molecules like this superimpose they do not line up directly above each other because areas of maximum density will align and repel. Instead they will slide a slight amount along the long and or short axis in order to adopt a packing mode that is more favorable. Halocyclophane molecules are hooked to each because of short alkyl chains that placed on the ends. This allows a cofacial configuration.
As the molecules approach each other there will be displacement along the long axis and the bandgap between HOMO an LUMO will change.
Influence of lateral displacements
At an intermolecular distance of 4 Å there will be an evolution of HOMO and LUMO according to the displacement of the two molecules. The graph begins with the gap of .2 and .25eV seen from the previous graph. The HOMO and LUMO have pronounced oscillations with increasing displacement. Notice that the level for HOMO goes from positive to negative passing through a zero where the electronic coupling and bandwidth for HOMO would be zero. The LUMO goes up and down but always has at least some bandwidth.
Phase difference between the evolutions of the �HOMO and LUMO splittings
Major fluctuations in bandwidths depend on the bonding/antibonding patterns of the HOMO/LUMO wavefunctions (the valence band and conduction bands). The diagram of polythiophene (PT) shows the bonding factor for the double bond at beta carbons in the HOMO and the antibonding factor at the single bond between the carbons (beneath the yellow sulfur) and between the rings.
Two molecules that are perfectly aligned have each positive lobe on the bottom molecule aligning with the positive lobe on the top molecule.
If you slide two LUMO molecules past each other the sign stays the same but the degree of overlap alternates with each shift. This results in the always positive fluctuating line in the graph. As the molecules slide off each other they have fewer rings that interact and the total splitting energy is less.
If you slide the HOMO molecules by half a ring the positive MO overlaps with a negative resulting in a negative splitting, a full ring displacement gives a positive with a positive and so on. Again the total bandgap steadily decreases as the molecules slide off each other and have few interacting MOs.
So as you displace molecules against each other there will be large variations in the overlap, and therefore changes in the electronic coupling. The HOMO and LUMO patterns are completely different and depend on their bonding and antibonding patterns.
Influence of lateral displacements along short axis
This graph shows the influence of lateral displacements along short axis which show large variations between HOMO an LUMO.
What counts is the wavefunction overlap NOT the spatial overlap. The bonding-antibonding pattern of the HOMO or LUMO wavefunction is the critical factor.
Quantum-chemistry calculations are required to predict this.
Molecular motions / vibrations have a major impact because they result in small movements which change the overlap.
Electron (conduction) bands can be intrinsically just as wide as hole (valence) bands. A misconception is that hole band must be narrower because of the number of nodes in antibonding orbitals in the LUMO. This logic doesn’t take into account overlap and displacements.
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