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 dat a analysis. Features such as vibrational frequency dispersion, nuclear equi librium displacement combined with vibrational frequency changes, and local mode anharmonicity are, however, important in many contexts but are not co vered by the simplest formalism. We provide here a new and simple parametri c scheme, with a single running parameter for inclusion of these effects in the diabatic and weakly adiabatic limits of electron transfer. The paramet er coincides with the symmetry factor in all cases of harmonic motion but w ith the local mode coordinate at the crossing of the potential surfaces whe n a local mode is significantly anharmonic. The scheme is numerical but eas y to use and does not rely on extensive computational efforts. The effects of vibrational frequency changes, nuclear tunnelling, and local mode anharm onicity are illustrated by calculation of free energy relations of electron transfer in homogeneous solution and current-overvoltage relations in elec trochemical electron transfer.