Reply:ET theory/Electronic coupling

[Silmar Andrade do Monte <silmar %-% at %-% tbi.univie.ac.at>]

	Dear Gavin,
 	For me it is always a pleasure to discuss about Electron Transfer.
 First, ET is a dynamic process, then should be treated by time-dependente
 quantum mechanics. As far as I know the easiest way to do this is by using
 the Fermi's Golden rule, which is a consequence of first-order time
 dependent perturbation theory.
 	In a two-state model, the first step is to identify the two
 relevant states for this model, i.e, the one with charge localized on the
 donor and the one with charge localized on the acceptor. These two states
 are called diabatic. The matrix element responsible for the transition
 between these two states is composed of a electronic coupling + terms
 concerning variation of diabatic states with nuclear coordinates(reaction
 coordinate). The most common electron transfer theories use the two
 following assumptions:
 (i) The electronic coupling(Tab) is >> than the other therms.
 (ii) Tab(Q)=Tab(Q*), i.e, the value of Tab is constant and is equal to the
 one obtained for the transition state (TS). This is de Franck-Condon
 factorization.
 	These two points are very well explained in an excellent Review by
 Patrick Bertrand(I don't have the reference now, but tomorrow i will send
 it to you).
 	I am not an expert in ET theory, but in my point of view the best
 way to understand it is by application in simple systems. Consider the
 molecule of H4+ with nuclear symmetry D2h(rectangle). The two diabatic
 states are the following:
 	|+  |   and  |   |+       For a reaction path with paralell
 aproximation between the
 	|   |        |   |
  the H2+ and H2 molecules this configuration corresponds to the TS. The
 question now is how to obtain the electronic coupling. These two states are
 non-orthogonal and can only be obtained by the method of localized
 orbitals(of Boys for example). The expression for Tab is
 		Tab = (Hab-HaaSab)/(1-S**2)
 In a normal calculation with Gaussian the SCF solution is a
 charge-delocalized one. This solution is called ADIABATIC and actually is a
 combination of the two charge-localized solutions. In order to obtain the
 charge-localized solutions it is necessary BREAKS THE SYMMETRY of the
 electronic wavefunction. Actually, these two solutions have
 symmetry(electronic) C2v. In practice these two solutions are only
 necessary if you want to obtain the matrix elements Hab and Sab. This can
 be done by the non-orthogonal CI method and the method of corresponding
 orbitals of Lowdin(I can give you the references if you want).
 For obtaining Tab are several levels of aproximation. In a one-electron
 level 2Tab corresponds to the difference E(HOMO)-E(HOMO-1). Increasing the
 accuracy you can perform a CI only with HOMO-1 -> HOMO excitation, and
 finally you can include more excitations in the CI, relaxing(MCSCF) or not
 the orbitals.
 	The exponential decay of Tab(with distance between the two H2
 molecules in the example) is a consequence of a the tunneling model, and
 spite its simple character is observed in many situations. Including a
 bridge between D and A often diminushes the rate of decay of Tab with
 distance, and there are some rules for the best bridges.
 	Finally for bigger, assymetric systems the major difficulty is to
 reach the transition state, but once reached these methods can be applied
 as well.
 	Well, that is all for the moment and we can continue this
 discussion anytime you want.
 		The best regards,
 							Silmar.