Reviews in Computational Chemistry Volume 18

170 Charge-Transfer Reactions in Condensed Phases
collective mode). In Eq. [65], both the linear coupling constant, Ci, and the
harmonic force constant, ki, change with the transition. The MH L model is
recovered when k1 ¼ k2. Note, that since the off-diagonal matrix elements of
the Hamiltonian are excluded from consideration, the formalism described
here may apply to any choice of wave functions for which such an approximation
is warranted. We therefore do not specify the basis set here, and the indices
i ¼ 1; 2 refer to any basis set in which the energies EiðqÞ are obtained.
The calculations of the diabatic (no off-diagonal matrix elements) free
energy surfaces in Eq. [18] can be performed exactly for EiðqÞ given by Eq.
[65]. This procedure yields the closed-form, analytical expressions for the
free energies FiðXÞ. It turns out that the solution exists only in a limited,
one-sided band of the energy gaps X. 61 Specifically, an asymptotic expansion
of the exact solution leads to a simple expression for the free energy
FiðXÞ ¼F0i þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jaijjX X0j
jaij
within a one-sided band of reaction coordinate X and
pffiffiffiffi 2
li
½66Š
FiðXÞ ¼1 ½67Š
outside the band.
The parameter X0 establishes the boundary of the energy gaps for which
a finite solution FiðXÞ exists. The band definition and its boundary
X0 ¼ I
C 2
2 k
both depend on the sign of the variation of the force constant k. The onesided
band is defined as (Figure 5):
fluctuation band ¼ X < X0 at k < 0
X > X0 at k > 0
F (X )
∆κ > 0
∞ ∞
∆κ < 0
X 0 X 0 X
Figure 5 Upper energy ( k > 0) and lower energy ( k < 0) fluctuations boundaries in
the Q model.
½68Š
½69Š