Branching and phase corrected surface hopping: A benchmark of nonadiabatic dynamics in multilevel systems

Since the seminal work of Tully [J. Chem. Phys. 93, 1061 (1990)], two-level scattering models have been extensively adopted as the standard benchmark systems to assess the performance of different trajectory surface hopping methods for nonadiabatic dynamics simulations. Here, we extend the branching and phase corrections to multilevel systems and combine them with both the traditional fewest switches surface hopping (FSSH) and its variant global flux surface hopping (GFSH) algorithms. To get a comprehensive evaluation of the proposed methods, we construct a series of more challenging and diverse three-level and four-level scattering models and use exact quantum solutions as references. Encouragingly, both FSSH and GFSH with the branching and phase corrections produce excellent and nearly identical results in all investigated systems, indicating that the new surface hopping methods are robust to describe multilevel problems and the reliability is insensitive to the definition of self-consistent hopping probabilities in the adiabatic representation. Furthermore, the branching correction is found to be especially important when dealing with strongly repulsive potential energy surfaces, which are common in realistic systems, thus promising for general applications.