Reviews in Computational Chemistry Volume 18

182 Charge-Transfer Reactions in Condensed Phases
intersection of the two curves (illustrated by circles in Figure 10) may occur.
Such a behavior was indeed observed in Ref. 78 for a series of porphyrin–
quinone diads in tetrahydrofuran. Maxima of the CS and CR curves get
closer to each other with decreasing solvent polarity, and, in fact, no curve
crossing was seen for the same systems in benzene as a solvent. 78 When the
normal region of CS is combined with the inverted region for CR, another scenario
is possible. The two branches (shown by squares in Figure 10) fitted by a
single curve (the dashed line in Figure 10) result in a plateau in the energy gap
law (a picture reminiscent of this behavior can be seen in Figure 4 of Ref. 79).
Nonlinear Solvation Effects
Experiment provides very limited evidence whether the free energy surfaces
of ET should be calculated invoking the linear or nonlinear solvent
response. In the absence of direct experimental evidence, the problem of
nonlinear solvation effects on the ET free energy surfaces has been approached
by computer simulations 33,51–53 and liquid-state solvation theories (integral
equations 80 and perturbation techniques 81 ). In computer simulations, the
free energy surfaces are calculated either directly by umbrella sampling
techniques 82 or indirectly by generating a few equilibrium cumulants. In
both cases, the lack of a general analytical framework to generate global
free energy surfaces from limited data available from simulations considerably
impedes the application of the simulation results to generate optical band
shapes or to make predictions concerning the ET energy gap law.
The Q model considered above may provide enough flexibility to be used
as an analytical background to analyze condensed-phase simulations of the ET
energetics. The great advantage of the model is that it requires only two first
equilibrium cumulants of the energy gap fluctuations for each electronic state
to generate FiðXÞ in the whole range of X values in the permissible fluctuation
band. The applicability of the model to mapping the simulations can be tested
on the consistency requirement given by Eq. [73]. Rewritten in terms of the
moments of the reaction coordinate X, this requirement implies that the factor
g ¼ hðdXÞ2 i 1
hðdXÞ 2 i 2
hðdXÞ 2 i2 þ 2kBT hXi
hðdXÞ 2 i1 þ 2kBT hXi
( hXi ¼hXi 2 hXi 1 ) should obey the condition
! 3
½102Š
g ¼ 1 ½103Š
Table 3 lists the parameters g extracted from simulations available in the
literature. The condition of Eq. [103] holds very well indeed, which allows one