Reviews in Computational Chemistry Volume 18

This result indicates a fundamental distinction between the Q and L models. In
the latter, the band of the energy gap fluctuations is not limited, leading to a
finite, even small, probability to find a fluctuation of any magnitude of the
energy gap. On the contrary, the Q model suggests a limited band for the
energy gap fluctuations. The gap magnitudes achievable due to the nuclear
fluctuations are limited by a low-energy boundary for k > 0 and by a
high-energy boundary for k < 0. The probability of finding an energy gap
fluctuation outside these boundaries is identically zero because there is no
real solution of the equation
X ¼ EðqÞ ¼E2ðqÞ E1ðqÞ ½70Š
The absence of a solution is the result of a bilinear dependence of the energy
gap EðqÞ on the driving nuclear mode q (Figure 6).
The other model parameters entering Eq. [66] are the nuclear reorganization
energies defined through the second cumulants of the reaction coordinate
li ¼ 1
2 bhðdXÞ2 i i ¼ 1
2ki ðCi=ai
and the relative changes in the force constants
ai ¼ ki
k
The two sets of parameters defined for each state are not independent because
of the following connections between them
∆E(q)
X 0
q
a 3 1 l1 ¼ a 3 2 l2
Beyond the Parabolas 171
CÞ 2
∆κ < 0 ∆κ > 0
Figure 6 The origin of the upper energy ( k < 0) and lower energy ( k > 0) fluctuation
boundaries due to a bilinear dependence of E on q.
X 0
q
½71Š
½72Š
½73Š