Subject: Reply:ET theory/Electronic coupling

From: Silmar Andrade do Monte <[email-masked]>
Date: Wed, 28 Feb 2001 21:32:48 +0100

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.