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.

先前的工作表明，通过使用量子力学的路径积分表示并将离散的电子状态映射到连续的笛卡尔变量，可以得出精确的量子“映射变量”环聚合物（MV-RP）哈密顿量。该MV-RP哈密顿量产生的经典分子动力学可用于精确计算多级量子系统的平衡性质，并近似实时热相关函数（TCF）。在这里，我们导出了MV-RP哈密顿量的混合时间切片形式，其中多级系统的不同模式被量化为不同程度。通过对一系列模型非绝热系统的量子实时TCF进行数值计算，我们探索了由这些哈密顿量产生的近似量子动力学的准确性，其中两个电子状态耦合到单个核自由度。有趣的是，我们发现由MV-RP哈密顿量产生的动力学经过经典处理的所有模式在此处考虑的所有模型系统中都比混合量化方法更稳定。此外，我们使用拉格朗日描述符来描述稳定极限和不稳定歧管，从而在经典极限MV-RP哈密顿量的6D相空间中表征非绝热动力学。