Simple schemes in chemical electron transfer formalism beyond single-mode quadratic forms: environmental vibrational dispersion and anharmonic nuclear motion
Simple quadratic rate constant forms, sometimes combined with a single high-frequency harmonic nuclear mode are commonly used in electron transfer data analysis. Features such as vibrational frequency dispersion, nuclear equilibrium displacement combined with vibrational frequency changes, and local mode anharmonicity are, however, important in many contexts but are not covered by the simplest formalism. We provide here a new and simple parametric scheme, with a single running parameter for inclusion of these effects in the diabatic and weakly adiabatic limits of electron transfer. The parameter coincides with the symmetry factor in all cases of harmonic motion but with the local mode coordinate at the crossing of the potential surfaces when a local mode is significantly anharmonic. The scheme is numerical but easy to use and does not rely on extensive computational efforts. The effects of vibrational frequency changes, nuclear tunnelling, and local mode anharmonicity are illustrated by calculation of free energy relations of electron transfer in homogeneous solution and current–overvoltage relations in electrochemical electron transfer.