Reviews in Computational Chemistry Volume 18

158 Charge-Transfer Reactions in Condensed Phases
In such cases, the instantaneous eigenvalues EiðqÞ of the solute electronic
Hamiltonian that form the free energy F elðqÞ
X
i
e bEiðqÞ ¼ e bF elðqÞ
should be considered as the basis for building the ET free energy surfaces. The
energies EiðqÞ can be used for nonequilibrium dynamics, since the population
of each surface is not limited by the condition of equilibrium as it is the case in
Eq. [16].
An electron is transferred between its centers of localization as a result of
underbarrier tunneling when the instantaneous electronic energies EiðqÞ come
into resonance due to thermal fluctuation or radiation of the medium
(Figure 1). The difference between the energies EiðqÞ thus makes a natural
choice for the ET reaction coordinate (cf. to Eq. [3])
½16Š
X ¼ EðqÞ ¼E2ðqÞ E1ðqÞ ½17Š
as first suggested by Lax 15 and then utilized in many ET studies. 5,17,32,33 The
reversible work necessary to achieve a particular magnitude of the energy gap
X defines the free energy profile of CT in terms of a Dirac delta function
e bFiðXÞþbF0i 1
¼ b Trn½dðXEðqÞÞe bEiðqÞ
Š=Trn½e bEiðqÞ
Š ½18Š
The partial trace in nuclear degrees of freedom in Eq. [13] is replaced in Eq.
[18] by the constraint imposed on the collective reaction coordinate X representing
the energy gap between the two levels involved in the transition. This
reduces the many-body problem of calculating the activation dynamics in the
coordinate space q to the dynamics over just one coordinate X. As we show in
the discussion of optical transition below, the same Boltzmann factor as in Eq.
[18] comes into expressions for optical profiles of CT bands. The solvent component
of the FCWD then becomes
FCWD s
i ðXÞ ¼be bFiðXÞþbF0i ½19Š
A more general definition of the FCWD includes overlap integrals of quantum
nuclear modes. 15,17 The definition given by Eq. [19] includes only classical solvent
modes (superscript ‘‘s’’) for which these overlap integrals are identically
equal to unity. An extension of Eq. [19] to the case of quantum intramolecular
excitations of the donor–acceptor complex is given below in the section
discussing optical Franck–Condon factors.