Ring polymer molecular dynamics with independent-bead approximation

It was recently reported [Zhao et al., Phys. Rev. Lett. 130, 166401 (2023)] that the full quantum dynamics of a correlated electron-nuclear system can be approximated to the dynamics of a classical ring polymer isomorphism with n beads each bearing distinct electronic configurations, namely, ring polymer molecular dynamics with independent-bead approximation. Here, we present a detailed description of the formalism, including the idea of this approximation and how it combines with existing Ehrenfest dynamics. This approach is applied to several model systems and compared with exact full quantum wavepacket dynamics, along with the widely used fewest switch surface hopping and standard Ehrenfest mean-field dynamics. The evolution of real-time electronic population and quantum nuclear trajectories obtained by this approach is in good agreement with the exact quantum solution, even in regions of strong non-adiabatic coupling, where the conventional surface hopping and Ehrenfest approaches fail to yield adequate results.