Mixed Quantum-Classical Study of Nonadiabatic Curve Crossing in Condensed Phases

We apply the mixed quantum-classical Liouville (MQCL) equation to investigate the nonadiabatic curve crossing in condensed phases. More specifically, electron transfer rate constants of the spin-Boson model are calculated by employing a rate constant expression using the collective solvent polarization as the reaction coordinate. In the calculation, classical nuclear degrees of freedom are initially sampled at the transition state configuration, and the initial state for the electronic degree of freedom is obtained from a mixed quantum-classical Boltzmann distribution. Different contributions to the electron transfer rate from the diagonal and off-diagonal elements of the initial density matrix, and contributions from trajectories with positive and negative initial velocities are analyzed. It is shown that the off-diagonal elements of the initial density matrix play an important role in the total electron transfer rate. The MQCL results are also compared with those calculated using Ehrenfest dynamics. It is found that, although the Ehrenfest dynamics is inaccurate when the reactive flux rate expression is used directly, it can give reasonably accurate results when individual contributions from the diagonal and off-diagonal elements of the initial density matrix are calculated.