Nonlinear quantum effects on electron transfer reactions

An approximate expression is developed for a nonadiabatic electron transfer rate to estimate quantum effects of nuclear rearrangements. The time-dependent formula for Fermiʹs golden rule is expanded by Plankʹs constant to the second order, using the Wigner transformation. The method of h̵-expansion is applicable to systems with nonlinear potentials or many degrees of freedom. Using a continuum approximation, from the expansion, a rate expression is obtained, including sizes of reactants and a distance between reactants explicitly. The ratio of the obtained rate to the classical rate agrees well with a ratio by a quantum Monte Carlo simulation.