Reviews in Computational Chemistry Volume 18

Paradigm of Free Energy Surfaces 157
achieved in treating the quantum dynamics in the framework of the functional
integral representation of the quantum subsystem. 25,26 In this approach,
the electron trace is taken out (TrelTr 0
n ) and is represented by a functional integral
over the quantum trajectories of the system. The inner trace over the
nuclear coordinates is then taken for each point of the quantum path by statistical
mechanics methods or by computer simulations of the many-particle
system.
The more traditional approach to treat the problem outlined by Eq. [13]
goes back to the theory of polarons in dielectric crystals. 27,28 It employs
the two-step procedure corresponding to two traces in Eq. [13]: first, the trace
over the electronic subsystem is taken with the subsequent restricted trace over
the nuclear coordinates. This approach, basic to the MH theory of ET, turns
out to be very convenient for a general description of several quantum dynamical
problems in condensed phases. It is currently widely used in steadystate
29 and time-resolved 2 spectroscopies and in theories of proton transfer, 30
dissociation reactions, 31 and other types of reactions in condensed media. The
central feature of the approach is the intuitively appealing and pictorially convenient
representation of the activated electron transition as dynamics on the
free energy surface of the reaction. Here, we start with outlining the basic steps
and concepts leading to the paradigm of the free energy surfaces. In this section,
we confine the discussion only to classical modes of the solvent. The
results obtained here are then used to discuss the construction of the
Franck–Condon factor of optical transitions, including quantum intramolecular
excitations of the donor–acceptor complex.
The first step of the derivation involves the BO approximation separating
the characteristic timescales of the electronic and nuclear motions in the
system. In this step, the instantaneous free energy depending on the system
nuclear coordinates q is defined by
e bF elðqÞ ¼ Trel e bH
For many homogeneous ET reactions, the energies of electronic excitations are
much higher than the energy of the thermal motion, which is of the order of
kBT. In such cases, the free energy F elðqÞ in Eq. [15] can be replaced by the
energy, independent of the bath temperature. This does not, however, happen
for electrochemical discharge where states of conduction electrons form a continuum
with thermal excitations between them. Entropic effects then gain
importance, and the free energy F elðqÞ should be considered in Eq. [15] (see
below).
The instantaneous free energy F elðqÞ is the equilibrium free energy,
implying equilibrium populations of the electronic states in the system. It is
not suitable for describing nonequilibrium processes with nonequilibrium
populations of the ground and excited states of the donor–acceptor complex.
½15Š