On the evaluation of the rate constant of the heterogeneous non-adiabatic electron transfer reaction

•General characteristics of the rate constant of the non-adiabatic heterogeneous electron transfer are considered.•A series of the extensions of the Marcus approximation is suggested and studied.•The new approximate expressions for the rate constant of the non-adiabatic heterogeneous outer-sphere electron transfer are obtained.•They have ≤ 1% error for all reaction free energies and realistic reorganization energies.

Some general characteristics of the rate constant of the non-adiabatic heterogeneous electron transfer are considered and the ways of the extensions of the Marcus approximation are studied. A number of the approximate formulas suggested earlier for the rate constant are examined critically with the emphasis on that obtained by Y. Zeng et al. [J. Electroanal. Chem. 735 (2014) 77–83]. We suggest the more accurate analytical formulas for the rate constant which can have applications in the practical calculations because they also permit avoiding the calculation of the improper integral over the Fermi distribution function. These approximate formulas are based on the integration by parts of the improper integral in the explicit expression for the rate constant without the use of any asymptotic matching and have ≤ 1% error for all reaction free energies and realistic reorganization energies.