Reviews in Computational Chemistry Volume 18

190 Charge-Transfer Reactions in Condensed Phases
Electronic transitions in the gas phase thus proceed from the lower state E1 to
the upper state E2. In condensed phases, these states are of course ‘‘dressed’’
by a solvating environment, but at FI s ¼ 0 one gets a nonzero equilibrium
driving force approximately equal to e E12 when E12=l I
1. The factor
e in the free energy driving force appears because the free energy represents
the work done to transfer the charge e ( z e at E12=l I
1, see
Figure 14) over the energy barrier E12 that results in e E12 for small splittings
E12.
Note above that the GMH 7 and adibatic formulations are equivalent in
terms of building the CT free energy surfaces. The distinctions seen in
Figure 15 may seem to contradict to this statement. The problem is resolved
by noting that the requirement Eab ¼ 0 imposed by the GMH formulation
makes the diabatic energy gap nonzero for self-exchange transitions:
I GMH
ab ¼ e E12 ½120Š
which is indeed the gap shown in Figure 15. The standard MH formulation 85
is then recovered when m12 ¼ 0 for symmetry reasons and thus e ¼ 1.
Nonlinear Solvation versus Intramolecular Effects
The origin of nonparabolic free energy surfaces of ET can be divided
into two broad categories: (1) intramolecular electronic effects and (2)
nonlinear solvation effects. Although these two origins can, at some instances,
be treated within the same mathematical framework (Q model), there are substantial
differences between them at both the quantitative and qualitative
levels. From the quantitative viewpoint, nonlinear solvation produces a
much weaker distortion of ET parabolas than do the polarizability change
and electronic delocalization. From a qualitative viewpoint, the two categories
of effects produce a nonzero nonparabolic distortion in different orders of the
expansion of the system Hamiltonian in the driving solvent mode.
The free energy FðPÞ invested in creation of a nonequilibrium solvent
polarization P can be expressed as a series in even powers of P with the two
first terms as follows:
FðPÞ ¼a1P 2 þ a2P 4
½121Š
where a1; a2 > 0. The interaction energy of the solute field with the solvent
polarization, U0s, is linear in P
U0s ¼ bP b > 0 ½122Š
For weak solute–solvent interactions, deviations from zero polarization of the
solvent are small, and one can keep only the first, harmonic, term in Eq. [121].