Simple formula for asymmetric Marcus–Hush kinetics

Abstract

The Marcus–Hush theory of electron transfer has seen increasing use as a predictive alternative to the phenomenological Butler–Volmer (BV) equation for Faradaic reactions kinetics. Here, we analyze and simplify the asymmetric Marcus–Hush (AMH) model, first proposed by Marcus and recently used by Compton’s group to fit experimental data that exhibit two different reorganization energies, depending on the sign of the overpotential. The AMH model has a single reorganization energy and an asymmetry parameter to account for different inner sphere force constants, but its practical use is hindered by the need to numerically evaluate the improper integral over the electronic Fermi distribution. Moreover, the domain of integration must be arbitrarily truncated to avoid divergence, due to some ambiguities in the derivation, which also limits the validity of the AMH model to weakly curved Tafel plots. Nevertheless, by defining a region over which the formula applies, we derive a simple formula to replace the Fermi integral by exploiting similarities with our recent approximation of the symmetric limit of the Marcus–Hush–Chidsey (MHC) model. These results enable the AMH model to approach the same ease of use as both the MHC and BV models and highlight the need to develop a more comprehensive theory of asymmetric charge transfer.