Reviews in Computational Chemistry Volume 18

186 Charge-Transfer Reactions in Condensed Phases
Here ‘‘inv’’ stands for an invariant in respect to transformation consistent with
the symmetry of the system. For quantum mechanical operators, this means
unitary transformations. The parameter e in Eq. [107] quantifies the extent
of mixing between two adiabatic gas-phase states induced by the interaction
with the solvent. For a dipolar solute, it is determined through the adiabatic
differential and the transition dipole moments
e ¼ 1 þ 4m2 12
m 2 12
1=2
½113Š
The differential and transition dipoles can be determined from experiment: the
former from the Stark spectroscopy 75,76 and the latter from absorption or
emission intensities (see below).
The parameter e should not be confused with the actual difference in
electronic occupation numbers of the two CT states. When the eigenfunctions
f ~ fþðY adÞ; ~ f ðYadÞg corresponding to the eigenstates F ðYadÞ are represented
as a linear combination of the wave functions of the adiabatic basis, ff1; f2g, ~f þðY ad ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Þ¼ 1 f ðYad q
ffiffiffiffiffiffiffiffiffiffiffiffiffi
Þf1
þ f ðYad q
Þf2
~f ðY ad ffiffiffiffiffiffiffiffiffiffiffiffiffi
Þ¼ f ðYad q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Þf1
þ 1 f ðYad q
½114Š
Þ
then the parameter f ðY ad Þ defines the occupation number of the adiabatic state
1 on the lower CT free energy surface at the reaction coordinate Y ad . For CT
transitions in the normal region, two equilibrium minima are located on the
lower CT free energy surface. The occupation number difference in the final
and initial states can thus be defined as
z ¼j1 f ðY 1 Þ f ðY 2 Þj ½115Š
where Y 1 and Y 2 are two minima positioned on the lower CT surface
(Figure 12). In contrast, when transitions between the lower and upper CT
surfaces occur in the inverted CT region, the occupation number difference
becomes
f 2
z ¼jf ðY þ Þ f ðY Þj ½116Š
where now Y þ and Y define the positions of equilibrium on the upper and
lower CT surfaces, respectively (Figure 13). Figure 14 illustrates the difference
in the dependence of the occupation number difference on e in the normal
and inverted CT regions. The parameter z is indeed close to e for reactions
with j F I s j lI . As the absolute value of the equilibrium energy gap increases,