Reviews in Computational Chemistry Volume 18

168 Charge-Transfer Reactions in Condensed Phases
thermal bath with the motion of a classical representative particle over the
activation barrier. 47 The MH solution gives the barrier as the vertical gap
between the bottom of the initial free energy surface and the intersection point.
The problem of finding the activation barrier then reduces to two parameters:
the free energy equilibrium gap, F0, and the classical nuclear reorganization
energy, l cl (Figure 2). From a broader perspective, as surprising as it seems, the
MH model for classical nuclear modes and its extension to quantum intramolecular
skeletal vibrations 17 presents the only exact, closed-form solution for
FiðXÞ available currently in the field of ET.
The success of the MH theory can also, to a large degree, be attributed to
the fact that the parameters of the model are connected to spectroscopic observables.
The first spectral moments for absorption and emission transitions
n abs=em fully define the classical reorganization energy l cl and the equilibrium
free energy gap F0 through the mean energy and the Stokes shift (Eqs. [6]
and [8])
h nst ¼ hðnabs nemÞ
¼ 2lcl ½64Š
Clearly, the MH description does not capture all possible complicated
mechanisms of ET activation in condensed phases. The general question
that arises in this connection is whether we are able to formulate an extension
of the mathematical MH framework that would (1) exactly derive from the
system Hamiltonian, (2) comply with the fundamental linear constraint in
Eq. [24], (3) give nonparabolic free energy surfaces and more flexibility to
include nonlinear electronic or solvation effects, and (4) provide an unambiguous
connection between the model parameters and spectroscopic observables.
In the next section, we present the bilinear coupling model (Q model),
which satisfies the above requirements and provides a generalization of the
MH model.
It has in fact been anticipated for many years that the CT free energy surfaces
may deviate from parabolas. A part of this interest is provoked by experimental
evidence from kinetics and spectroscopy. First, the dependence of the
activation free energy, Fact i , for the forward (i ¼ 1 ) and backward (i ¼ 2) reactions
on the equilibrium free energy gap F0 (ET energy gap law) is rarely a
symmetric parabola as is suggested by the Marcus equation, 48 Eq. [9]. Second,
optical spectra are asymmetric in most cases 17 and in some cases do not show
the mirror symmetry between absorption and emission. 49 In both types of
experiments, however, the observed effect is an ill-defined mixture of the intramolecular
vibrational excitations of the solute and thermal fluctuations of the
solvent. The band shape analysis of optical lines does not currently allow an
unambiguous separation of these two effects, and there is insufficient information
about the solvent-induced free energy profiles of ET.
Nonlinear solvation (breakdown of assumption 4 in the Introduction)
has long been considered as the main possible origin of nonparabolic free