A ring polymer molecular dynamics extension of the mapping approach to surface hopping: Incorporation of nuclear quantum effects

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In this work, we aim to incorporate explicit nuclear quantum effects into the mapping approach to surface hopping (MASH). To accomplish this goal, we develop an extension of MASH using the ring-polymer (RP) formalism, which we term RP-MASH. Within RP-MASH, the nonadiabatic electronic transitions are described with explicit equations of motion using electronic mapping variables, while the nuclei are quantized through the RP Hamiltonian. The final equations of motion are fully deterministic, consistent with the original MASH formalism, and the entire RP follows a single adiabatic potential energy surface at all times. An electronic transition leads to all beads switching simultaneously to the new potential energy surface. We define two versions of RP-MASH, distinguished by whether the terms involving the nonadiabatic coupling vector are calculated as a bead-average over the positions of all beads or from the position of the centroid of the RP. We apply RP-MASH to Tully's avoided crossing models, internal conversion in the 3-mode model for pyrazine, and the spin-boson model for electron transfer. We find that MASH and RP-MASH perform similarly in the first two models when the initial state of the system is treated appropriately. In comparison, the electron transfer rate constants for the spin-boson model calculated using RP-MASH show a significant improvement over MASH. Taken together, the results indicate that explicit nuclear quantum effects are most relevant for thermally driven processes and that RP-MASH is a practical and useful simulation methodology to tackle such dynamics.

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