On the evaluation of the Marcus-Hush-Chidsey integral

The electrochemical rate constant obtained from the Marcus-Hush theory of heterogeneous electron transfer is given as a relatively complex integral. Recently, two apparently different expressions of this rate constant in the form of a series of analytical functions appeared in the literature. We demonstrate here the equivalence of these expressions and discuss their different approximations, resulting from the two distinct analytical derivations, which have implications in the practical calculation of electron transfer rate constants at electrode surfaces.