Reviews in Computational Chemistry Volume 18

156 Charge-Transfer Reactions in Condensed Phases
The free energy gap, equal to the energy of the incident light, is basis independent.
It defines the Franck–Condon factor entering the optical band shapes.
The analysis below follows this general scheme (Figure 4).
Formulation
Electron transfer and, more broadly, CT reactions belong to a general
class of problems having a quantum subsystem interacting with a condensed-phase
thermal bath. The main challenge in describing such systems is
the necessity to treat the quantum subsystem coupled to a wide spectrum of
classical and quantum modes of the condensed environment. It implies that
the calculation of some property of interest F involves taking a restricted statistical
average (trace, Tr) over both the electronic and nuclear modes
where
Hamiltonian
Statistical average over
the electronic degrees of freedom
F el (q)
Projection on
the nuclear polarization Pn Diabatic Y
Projection on the energy gap
reaction coordinate, X
d Y ad
Franck−Condon factor
FðQ; tÞ ¼Tr 0
nTrel½^rðtÞŠ ½13Š
^rðtÞ ¼e iHt ^rð0Þe iHt
Adiabatic
Figure 4 Hierarchy of reaction coordinates in deriving the Franck–Condon factor from
the system Hamiltonian.
is the density matrix of the system defined by the Hamiltonian H; ^rð0Þ ¼
expð bHÞ and b ¼ 1=kBT. The quantity Trel denotes the trace over the electronic
degrees of freedom, and Tr 0
n refers to an incomplete or restricted trace
over the nuclear degrees of freedom, excluding a manifold of modes Q that are
of interest for some particular problem.
Depending on the order of the statistical average in Eq. [13], there are
two basic approaches to calculate FðQ; tÞ. Considerable progress has been
½14Š