Reviews in Computational Chemistry Volume 18

184 Charge-Transfer Reactions in Condensed Phases
in Figure 11 shows the dependence of a1 on the magnitude of solute’s dipole.
The dipolar lattice demonstrates a considerably higher extent of nonlinear solvation
compared to dipolar liquids. The reason for this effect is that the lattice
dipoles are immobilized and the orientational saturation is not compensated by
local density compression as happens in liquid solvents. 83
Electron Delocalization Effects
Equations [41]–[42] give a general, exact solution for the free energy surfaces
of a two-level system characterized by two collinear vectors: differential,
E ab, and off-diagonal, E ab, electric fields of the donor–acceptor complex.
When the off-diagonal matrix elements are nonnegligible, the free energy surfaces
are substantially nonparabolic. They are defined by five parameters: l d ,
F d s , I ab, H ab, anda ab. A careful choice of the basis set allows the elimination
of one parameter. Two approaches can be employed. In the adiabatic basis
set, ff 1; f 2g, the gas-phase ET matrix element is zero, H12 ¼ 0. Alternatively,
one can define the basis set by demanding the off-diagonal matrix element of
the solute electric field be zero, a ab ¼ 0. This choice sets up the generalized
Mulliken–Hush (GMH) basis. 7 These two approaches are essentially equivalent
in terms of building the CT free energy surfaces, 42 but the adiabatic basis
may be more convenient for practical applications. The reason is that most
quantum chemical software packages are designed to diagonalize the gasphase
Hamiltonian matrix, thus generating the adiabatic basis and corresponding
adiabatic matrix elements of the solute electric field.
There are several fundamental reasons why the GMH and adiabatic formulations
are to be preferred over the traditionally employed diabatic formulation.
The definition of the diabatic basis set is straightforward for
intermolecular ET reactions when the donor and acceptor units are separated
before the reaction and form a donor–acceptor complex in the course of diffusion
in a liquid solvent. The diabatic states are then defined as those of separate
donor and acceptor units. The current trend in experimental design of
donor–acceptor systems, however, has focused more attention on intramolecular
reactions where the donor and acceptor units are coupled in one molecule
by a bridge. 22 The direct donor–acceptor overlap and the mixing to bridge
states both lead to electronic delocalization, 75,76 with the result that the centers
of electronic localization and localized diabatic states are ill-defined. It is
then more appropriate to use either the GMH or adiabatic formulation.
There is an additional, more fundamental, issue involved in applying the
standard diabatic formalism. The solvent reorganization energy and the solvent
component of the equilibrium free energy gap are bilinear forms of
E ab and Eav (Eqs. [45] and [47]). A unitary transformation of the diabatic
basis (Eq. [27]), which should not affect any physical observables, then
changes E ab and Eav, affecting the reorganization parameters. The activation
parameters of ET consequently depend on transformations of the basis set!