Influence of distortion and Duschinsky effects on Marcus-type theories of electron transfer rate

Abstract

In this paper, we focus on the microscopic theory of intra-molecular electron transfer (ET) rate. Specifically, we examine whether or not and/or under what conditions the widely-used Marcus-type equations are applicable to displaced–distorted (D-D) and displaced–distorted-rotated (D-D-R) harmonic oscillator (HO) cases. For this purpose, we apply the cumulant expansion (CE) method to derive the ET rate constants for these cases. Within the CE method, we find the analytical condition upon which the Marcus-type equations of the Gaussian form can be obtained for the D-D HO case as well as the displaced HO case. If there is significant distortion or Duschinsky mixing (mode mixing), the Marcus-type equations of the Gaussian form are not adequate, and we discuss the reasons for the breakdown of this form. We also find that the reorganization energy and the free energy change for the D-D HO depend on the temperature. This temperature dependent feature is different from the displaced HO case in which the reorganization energy and the free energy change are independent of the temperature. As a consequence, the pre-exponential factor of the ET rate shows a temperature dependence different from the usual 1/√T behavior. We explain this analytically, and show the effect by several numerical examples.