On Quantum, Classical and Semiclassical Calculations of Electron Transfer Rates

Similarities and differences between quantum, semiclassical and classical treatments of electron transfer reactions are described. All of these methods can be described as "vibrationally assisted electron-tunneling." A simple approximate expression is given for an infinite sum of Bessel functions which appears in the nonadiabatic quantum description. The condition of its validity is also obtained. The expression has been tested for some systems when the frequencies contain those in the range of the usual metal - ligand frequencies (~450 cm^(-1)> and of the low frequencies of the environment. The nonadiabatic classical expression provides a useful lower bound, because it has no nuclear tunneling, to the nonadiabatic quantum result. The quantum correction is calculated for some actual systems. The "semiclassical" approximation is sometimes larger and sometimes smaller (in the "normal" and "inverted" regions, respectively) than the quantum result. This behavior is readily understood in terms of WKB theory. The quantum correction to the classical cross-relation of rate constants was calculated for a case of interest and found to be minor. A symmetric form of (lnk, Δ^0) plots is described, even for the usually skewed quantum plots