Semiclassical electron transfer: Zusman equations versus Langevin approach Original Research Article

The model of a donor–acceptor electron transfer (ET) being coupled to an overdamped reaction coordinate is studied semiclassically. Starting from an archetypal ET Hamiltonian, two different routes of the semiclassical approximation are critically analyzed and mutually compared. A first path proceeds along the Caldeira–Leggett form for the master equation for the reduced density matrix of “electron+reaction coordinate” which is cast in its Wigner phase–space representation. Integrating over the momentum of the reaction coordinate then yields (in the overdamped limit) the so-termed Zusman ET-equations. The alternative route starts from the formally exact quantum Heisenberg–Langevin equations. The corresponding derivation of the equations of motion for the observables involves a sort of a mean-field semiclassical approximation. The final result are nonlinear stochastic differential equations (SDE) of the Langevin type. We compare the results of these two approaches, both in the absence and in the presence of external time-dependent driving fields. Our findings are that both methods yield good agreement for the description of symmetric ET. In contrast, however, the SDE fails to describe the ET dynamics when a strong static bias is present. The inclusion of a strong time-periodic field ET-manipulation improves the Langevin approximation scheme, providing reasonable good agreement between both routes.