Surface hopping with nuclear quantum effects through path-integral coarse graining

Incorporating nuclear quantum effects (NQEs), such as tunneling and zero-point energy, in nonadiabatic dynamics is of fundamental importance. We propose a simple and robust scheme based on introducing an effective quantum potential for the nuclei in mixed quantum-classical trajectory approaches. The most straightforward method is to combine the fewest-switches surface hopping (FSSH) method with centroid molecular dynamics, which we name as centroid-surface surface hopping. Furthermore, the unfavorable computational overhead of path-integral simulations can be mitigated via the recently developed path-integral coarse-graining scheme, achieving efficiency comparable to FSSH with classical nuclei. The method is benchmarked against exact quantum mechanical calculations on two one-dimensional two-state model systems, each with three sets of parameters covering both the golden-rule and near-adiabatic regimes. Results show that our method predicts quantitatively accurate reaction rates and correlation functions, even in the deep tunneling regime, on both model systems for all the parameter sets tested. This development paves the way for efficient and accurate modeling of NQEs in nonadiabatic reactions/processes in the future.