Reviews in Computational Chemistry Volume 18

162 Charge-Transfer Reactions in Condensed Phases
Eq. [33] according to the assumption of the classical character of this collective
mode. Depending on the form of the coupling of the electron donor–acceptor
subsystem to the solvent field, one may consider linear or nonlinear solvation
models. The coupling term Ei P in Eq. [32] represents the linear coupling
model (L model) that results in a widely used linear response approximation. 37
Some general properties of the bilinear coupling (Q model) are discussed
below.
Equations [32] and [33] represent the system Hamiltonian that can be
used to build the CT free energy surfaces. According to the general scheme
outlined above, the first step in this procedure is to take the average over
the electronic degrees of freedom of the system. This implies integrating
over the electronic polarization Pe and the fermionic populations a þ i ai. The
trace Tr el can be taken exactly, resulting in two instantaneous energies 38
E ½PnŠ ¼ ~ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 Iav½PnŠ 2 ð ~ I½PnŠÞ 2 þ 4ðHeff q
2
ab ½PnŠÞ
where ~ Iav ¼ð ~ Ia þ ~ I bÞ=2 and ~ I ¼ ~ Ib
~ Ii½PnŠ ¼Ii Ei Pn
~ Ia. For i ¼ a; b
The effective ET matrix element has the form
½35Š
1
2 Ei ð we Ei þE12 we E12Þ
½36Š
~H eff
ab ½PnŠ ¼e Se=2
½Hab Eab Pn Eav we EabŠ ½37Š
with Eav ¼ðEa þE bÞ=2. The matrix element ~ H eff
ab ½PnŠ depends on the solvent
through two components: (1) interaction of the off-diagonal solute electric
field with the nuclear solvent polarization (second term) and (2) solvation of
the off-diagonal field by the electronic polarization of the solvent (third term).
The former component leads to solvent-induced fluctuations of the ET matrix
element, which represent a non-Condon effect 39 of the dependence of electron
coupling on nuclear degrees of freedom of the system. This effect is commonly
neglected in the Condon approximation employed in treating nonadiabatic ET
rates. 11
Equation [37] is derived within the assumption that both the electronic
polarization and the donor–acceptor complex are characterized by quantum
excitation frequencies, 38 bhoe 1, b E12 1, where E12 ¼ E2 E1 is
the gas-phase adiabatic energy gap in Eq. [29]. The derivation does not assume
any particular separation of these two characteristic time scales. The traditional
formulation 27 assumes E12 hoe that eliminates the electronic
Franck–Condon factor expð Se=2Þ in Eq. [37]. The parameter 38,40
Se ¼ E ab w e E ab=2hoe E ab ¼E b Ea ½38Š