  Recently proposed non-adiabatic ring polymer molecular dynamics (NRPMD)
approach has shown to provide accurate quantum dynamics by incorporating
explicit electronic state descriptions and nuclear quantizations. Here, we
present a rigorous derivation of the NRPMD Hamiltonian and investigate its
performance on simulating excited state non-adiabatic dynamics. Our derivation
is based on the Meyer-Miller-Stock-Thoss (MMST) mapping representation for
electronic states and the ring-polymer path-integral description for nuclei,
resulting in the same Hamiltonian proposed in the original NRPMD approach. In
addition, we investigate the accuracy of using NRPMD to simulate photoinduced
non-adiabatic dynamics in simple model systems. These model calculations
suggest that NRPMD can alleviate the zero-point energy leakage problem that is
commonly encountered in the classical Wigner dynamics, and provide accurate
excited states non-adiabatic dynamics. This work provides a solid theoretical
foundation of the promising NRPMD Hamiltonian and demonstrates the possibility
of using state-dependent RPMD approach to accurately simulate electronic
non-adiabatic dynamics while explicitly quantize nuclei.
