Reviews in Computational Chemistry Volume 18

172 Charge-Transfer Reactions in Condensed Phases
and
a2 ¼ 1 þ a1
An additional constraint on the magnitudes of the parameter a1 comes from
the condition of the thermodynamic stability of the collective solvent mode in
both states, ki > 0, resulting in two inequalities
½74Š
a1 > 0 or a1 < 1 ½75Š
The inequalities in Eq. [75] also define the condition for the generating function
(Eq. [23]) to be analytic in the integration contour in Eq. [25]. This condition
is equivalent to the linear connection between the diabatic free energy
surfaces, Eq. [24]. The Q model solution thus explicitly indicates that the linear
relation between the diabatic free energy surfaces is equivalent to the condition
of thermodynamic stability of the collective nuclear mode driving ET.
Equations [73] and [74] reduce the number of independent parameters of
the Q model to three: F0, l1, and a1. Here, F0 (Eq. [21]) is the free energy
gap between equilibrium configurations of the system (Figure 2). The fluctuation
boundary X0 is connected to F0 by the relation
X0 ¼ F0 þ l1a 2 1 =a2 ½76Š
Compared to the two-parameter MH theory (l and F0), 12 the Q model introduces
an additional flexibility in terms of the relative variation of the fluctuation
force constant through a1. The MH theory is recovered in the limit
a1 !1.
Importantly, the new free energy surfaces lead to qualitatively new
features for the activated ET kinetics. The standard high-temperature limit
of two diabatic ET free energy surfaces
FiðXÞ ¼F0i þ ðX F0 liÞ 2
is reproduced when ai 1 (the driving mode force constants ki in the two
states are similar) and, additionally, jX F0 lij jaijli. Here, ‘‘ ’’ and
‘‘þ’’ correspond to i ¼ 1 and i ¼ 2, respectively. The second requirement
implies that the reaction coordinate should be not too far from the free energy
minimum to preserve its parabolic form. By contrast, in the limit
jX X0j lijaij, the linear dependence wins over the parabolic law
4li
a
FiðXÞ ¼F0i þjaij X F0 þ l1
2 1
a2
½77Š
½78Š