Detailed balance, the correct thermalization of electronic state populations, is an essential and elusive property of quantum–classical non-adiabatic dynamics methods. While some methods can reproduce detailed balance through physically well-motivated algorithmic adaptations, or by construction of a conserved Hamiltonian function, the physical mechanism leading to detailed balance is not understood from first principles. Coupled trajectory mixed quantum–classical (CTMQC) dynamics may provide some insight into the question, as it can be derived from first principles in the exact factorization theorem of full quantum mechanics. Although we find that the current conventional flavor of CTMQC, which conserves energy across the ensemble of trajectories (known as CTMQC-E), fails to reproduce detailed balance as in Ehrenfest dynamics, we show that a similar variant, where total energy is conserved on each trajectory independently, provides a major improvement over Ehrenfest with respect to detailed balance. Moreover, we show that the theory achieves convergence of the mean electronic potential energy with the number of energy levels that successively increase in energy. This new variant is shown to, by simulations on the Tully models and double arch model, retain a good description electronic populations and coherence compared to exact quantum dynamics. We explain the thermalization mechanism through the additional terms that distinguish CTMQC from Ehrenfest dynamics. We show that the improvement can be explained via geometric contributions to the nuclear force, resulting from the quantum momentum, which act to oppose motion when electrons decohere upward in energy and act to enhance motion otherwise, somewhat emulating the mechanism of frustrated hops. These results have considerable implications for the applicability of CTMQC to condensed phase simulations.
Dines et al. (Mon,) studied this question.