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We develop a density matrix formalism to describe coupled electron-nucleardynamics. To this end we introduce an effective Hamiltonian formalism thatdescribes electronic transitions and small (quantum) nuclear fluctuations alonga classical trajectory of the nuclei. Using this Hamiltonian we deriveequations of motion for the electronic occupation numbers and for the nuclearcoordinates and momenta. We show that in the limit when the number of nucleardegrees of freedom coupled to a given electronic transition is sufficientlyhigh (i.e., the strong decoherence limit), the equations of motion for theelectronic occupation numbers become Markovian. Furthermore the transitionrates in these (rate) equations are asymmetric with respect to thelower-to-higher energy transitions and vice versa. In thermal equilibrium suchasymmetry corresponds to the detailed balance condition. We also study theequations for the electronic occupations in non-Markovian regime and develop asurface hopping algorithm based on our formalism. To treat the decoherenceeffects we introduce additional "virtual" nuclear wavepackets whoseinterference with the "real" (physical) wavepackets leads to the reduction incoupling between the electronic states (i.e., decoherence) as well as to thephase shifts that improve the accuracy of the numerical approach. Remarkably,the same phase shifts lead to the detailed balance condition in the strongdecoherence limit.

Mixed quantum-classical approach to model non-adiabatic electron-nuclear dynamics: Detailed balance and improved surface hopping method