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Semiclassical approaches are widely employed for understanding nonadiabatic processes in complex systems. However, many semiclassical approaches may suffer from various unphysical behaviors especially in the long time limit. For example, the Poisson bracket mapping equation (PBME), an example of semiclassical approaches that can be usefully adopted in simulating large systems, sometimes displays negative populations in long simulations. Here, to reduce the error in such population dynamics, we present a mapping variable scaling approach for PBME. We demonstrate that our approach yields the equilibrium population reliably in the long time limit by simulating energy transfers in a series of model systems. Based on error analyses of the system density matrices, we determine conditions for reliable dynamics in model two-state systems. We then apply our scheme to following the energy transfer dynamics in a more realistic seven state model with parameters that reflect experimental situations. With this, we confirm that the modified PBME provides correct equilibrium populations in the long time limit, with acceptable deterioration in the short time dynamics. We also test how the initial bath energy distribution changes in time depending on the schemes of sampling the initial bath modes, and try to see its effect on the system dynamics. Finally, we discuss the applicability of our scaling scheme to all-atom style semiclassical simulations of complex systems.

Improving long time behavior of Poisson bracket mapping equation: A mapping variable scaling approach