Nonadiabatic anharmonic electron transfer

The effect of an inner sphere, local mode vibration on an electron transfer is modeled using the nonadiabatic transition probability (rate) expression together with both the anharmonic Morse and the harmonic oscillator potential. For an anharmonic inner sphere mode, a variational analysis uses harmonic oscillator basis functions to overcome the difficulties evaluating Morse-model Franck-Condon overlap factors. Individual matrix elements are computed with the use of new, fast, robust, and flexible recurrence relations. The analysis therefore readily addresses changes in frequency and/or displacement of oscillator minimums in the different electron transfer states. Direct summation of the individual Boltzmann weighted Franck-Condon contributions avoids the limitations inherent in the use of the familiar high-temperature, gaussian form of the rate constant. The effect of harmonic versus anharmonic inner sphere modes on the electron transfer is readily seen, especially in the exoergic, inverted region. The behavior of the transition probability can also be displayed as a surface for all temperatures and values of the driving force/exoergicity Δ = -ΔG. The temperature insensitivity of the transfer rate is clearly seen when the exoergicity equals the collective reorganization energy (Δ = Λ(s)) along a maximum ln (w) vs. Δ ridge of the surface. The surface also reveals additional regions for Δ where ln (w) appears to be insensitive to temperature, or effectively activationless, for some kinds of inner sphere contributions.