Reviews in Computational Chemistry Volume 18

198 Charge-Transfer Reactions in Condensed Phases
with the Mulliken–Hush 6 ET matrix element
H MH ¼ H ab
ðm ab
m 2 ab
m abÞ
I ab
½141Š
where m ab and m ab refer to, respectively, the gas-phase transition and differential
dipole moments calculated in the diabatic basis set; I ab is the diabatic
gas-phase energy gap. The term Mulliken–Hush 6 here refers to the fact that the
matrix element in Eq. [141] is related to the projection of the adiabatic transition
dipole on the direction of the difference diabatic dipole
H MH ¼ ðm12 mabÞ m2 ab
E12
½142Š
Under the special condition that m12 and m ab are parallel, one obtains the
MH relation 6,7
H MH ¼ m12
E12
½143Š
mab Equations [140]–[143] provide a connection between the preexponential factor
entering the nonadiabatic ET rate and the spectroscopically measured adiabatic
transition dipole m12. It turns out that the Mulliken–Hush matrix
element, commonly considered as an approximation valid for mab ¼ 0, 7 enters
exactly the rate constant preexponent as long as the non-Condon
solvent effects are accurately taken into account. 43 Equation [142] stresses
the importance of the orientation of the adiabatic transition dipole relative
to the direction of ET set up by the difference diabatic dipole mab. The value
of HMH is zero when the vectors m12 and mab are perpendicular.
Electronically Delocalized Chromophores
Equation [48] gives the Franck–Condon factor that defines the probability
of finding a system configuration with a given magnitude of the energy gap
between the upper and lower CT free energy surfaces. It can be directly used to
define the solvent band shape function 96 of a CT optical transition in Eq. [134]
where
FCWD s ðn mhnvÞ ¼Q
Q ¼
1 X
k¼1;2
E 0 j ðYkmÞj 1 exp½ bF ðYkmÞŠ ½144Š
ð
e bF ðYad Þ dY ad
½145Š