Investigating the Stability and Accuracy of a Classical Mapping Variable Hamiltonian for Nonadiabatic Quantum Dynamics

Previous work has shown that by using the path integral representation of quantum mechanics and by mapping discrete electronic states to continuous Cartesian variables, it is possible to derive an exact quantum “mapping variable” ring-polymer (MV-RP) Hamiltonian. The classical molecular dynamics generated by this MV-RP Hamiltonian can be used to calculate equilibrium properties of multi-level quantum systems exactly, and to approximate real-time thermal correlation functions (TCFs). Here, we derive mixed time-slicing forms of the MV-RP Hamiltonian where different modes of a multi-level system are quantized to different extents. We explore the accuracy of the approximate quantum dynamics generated by these Hamiltonians through numerical calculation of quantum real-time TCFs for a range of model nonadiabatic systems, where two electronic states are coupled to a single nuclear degree of freedom. Interestingly, we find that the dynamics generated by an MV-RP Hamiltonian with all modes treated classically is more stable across all model systems considered here than mixed quantization approaches. Further, we characterize nonadiabatic dynamics in the 6D phase space of our classical-limit MV-RP Hamiltonian using Lagrangian descriptors to identify stable and unstable manifolds.