Simple formula for Marcus–Hush–Chidsey kinetics

• A simple formula is derived to approximate the Marcus–Hush–Chidsey integral.
• It asymptotically matches exact limiting expressions.
• It has <5% error for all overpotentials and realistic reorganization energies.
• It can be evaluated almost as quickly as the Butler–Volmer equation.

The Marcus–Hush–Chidsey (MHC) model is well known in electro-analytical chemistry as a successful microscopic theory of outer-sphere electron transfer at metal electrodes, but it is unfamiliar and rarely used in electrochemical engineering. One reason may be the difficulty of evaluating the MHC reaction rate, which is defined as an improper integral of the Marcus rate over the Fermi distribution of electron energies. Here, we report a simple analytical approximation of the MHC integral that interpolates between exact asymptotic limits for large overpotentials, as well as for large or small reorganization energies, and exhibits less than 5% relative error for all reasonable parameter values. This result enables the MHC model to be considered as a practical alternative to the ubiquitous Butler–Volmer equation for improved understanding and engineering of electrochemical systems.