Failure of the Born–Oppenheimer and Franck–Condon approximations for long distance electron transfer rate calculations

Quantum mechanical and semiclassical formulations of nonadiabatic electron transfer theory are usually implemented within a Born–Oppenheimer regime. Calculations on real weakly interacting systems are so difficult that this approximation is rarely questioned. The Born–Oppenheimer approximation becomes qualitatively wrong for electron transfers at very large distances. A model vibronic problem is exactly solved and compared with the Born–Oppenheimer result. Rate expressions are derived from the wave functions using the ‘‘golden rule’’ approximation. Electron propagation is intimately correlated to nuclear motion so that the vibrational energy left on the donor critically affects the electronic decay length. Several deviations from the usual predictions appear for transfers over very large distances.