Reviews in Computational Chemistry Volume 18

Paradigm of Free Energy Surfaces 155
donor–acceptor complex. A description of CT activation and spectroscopy in
terms of two crossing, free energy surfaces (Figure 2) is in fact possible for any
choice of the basis set as long as the off-diagonal matrix elements of the solute
quantum mechanical operators can be neglected. In cases when a description
in both diabatic and adiabatic representations is possible (as it is for the
Q-model discussed below), we will not specify the basis by dropping the
‘‘d’’ and ‘‘ad’’ superscripts.
The statistical mechanical analysis of ET and CT free energy surfaces
developed in the first part of this chapter is applied to the calculation of optical
absorption and emission profiles in the second part. This application of the
theory, related to the band shape analysis of optical line shapes, has been a central
issue in understanding CT energetics for several decades. 16 The chapter is
designed to demonstrate how the extension of the basic models used to
describe the thermodynamics of CT is reflected in asymmetry of the energy
gap law (dependence of the CT activation barrier on the equilibrium free energy
gap) and more complex and structured optical band shapes. The development
of a corresponding band shape analysis incorporating these new features is in
its infancy, and we will certainly see more activity in this field in the future.
PARADIGM OF FREE ENERGY SURFACES
The CT/ET free energy surface is the central concept in the theory of CT/
ET reactions. The surface’s main purpose is to reduce the many-body problem
of a localized electron in a condensed-phase environment to a few collective
reaction coordinates affecting the electronic energy levels. This idea is based
on the Born–Oppenheimer (BO) separation 24 of the electronic and nuclear
time scales, which in turn makes the nuclear dynamics responsible for fluctuations
of electronic energy levels (Figure 1). The choice of a particular collective
mode is dictated by the problem considered. One reaction coordinate stands
out above all others, however, and is the energy gap between the two CT states
as probed by optical spectroscopy (i.e., an experimental observable).
Our discussion of the CT free energy surfaces involves a hierarchy of
reaction coordinates (Figure 4). It starts from the instantaneous free energy
surfaces obtained from tracing out (statistical averaging) the electronic degrees
of freedom in the system density matrix (i.e., solving the electronic problem
for fixed nuclear coordinates). In the case when the direction of electron transfer
sets up the only preferential direction in the CT system, one can define a
scalar reaction coordinate as the projection of the nuclear solvent polarization
on the differential electrical field of the solute. Depending on the basis set
employed, this gives the diabatic or adiabatic scalar reaction coordinates, Y d
and Y ad (Figure 4). At this step, a reaction coordinate depends on the basis set
of solute wave functions employed. This dependence is eliminated when a scalar
reaction coordinate is projected on the energy gap between the CT surfaces.