We present two multistate ring polymer instanton (RPI) formulations, both obtained from an exact path integral representation of the quantum canonical partition function for multistate systems. The two RPIs differ in their treatment of the electronic degrees of freedom; while the Mean-Field (MF)-RPI averages over the electronic state contributions, the Mapping Variable (MV)-RPI employs explicit continuous Cartesian variables to represent the electronic states. We compute both RPIs for a series of model two-state systems coupled to a single nuclear mode with electronic coupling values chosen to describe dynamics in both adiabatic and nonadiabatic regimes. We show that the MF-RPIs for symmetric systems are in good agreement with the previous literature, and we show that our numerical techniques are robust for systems with non-zero driving force. The nuclear MF-RPI and the nuclear MV-RPI are similar, but the MV-RPI uniquely reports on the changes in the electronic state populations along the instanton path. In both cases, we analytically demonstrate the existence of a zero-mode, and we numerically find that these solutions are true instantons with a single unstable mode as expected for a first order saddle point. Finally, we use the MF-RPI to accurately calculate rate constants for adiabatic and nonadiabatic model systems with the coupling strength varying over three orders of magnitude.

中文翻译：

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多态环聚合物实例和非绝热反应速率

我们提出了两种多态环聚合物瞬时子（RPI）公式，它们都从多态系统的量子规范分配函数的精确路径积分表示中获得。这两个RPI在处理电子自由度方面有所不同。虽然平均场（MF）-RPI对电子状态贡献进行平均，但映射变量（MV）-RPI使用显式连续笛卡尔变量表示电子状态。我们计算了一系列耦合到单核模式的模型两态系统的两个RPI，并选择了电子耦合值来描述绝热和非绝热状态下的动力学。我们证明了对称系统的MF-RPI与以前的文献非常吻合，并且我们证明了我们的数值技术对于非零驱动力的系统是鲁棒的。核MF-RPI和核MV-RPI相似，但MV-RPI唯一报告沿瞬时路径的电子态总体变化。在这两种情况下，我们都通过分析证明了零模式的存在，并且在数值上发现这些解决方案是真实的具有单个不稳定模式的瞬时子，这是一阶鞍点所期望的。最后，我们使用MF-RPI精确计算了绝热和非绝热模型系统的速率常数，其耦合强度变化了三个数量级。并且我们从数字上发现这些解决方案是真实的实例，具有一阶鞍点所期望的单个不稳定模式。最后，我们使用MF-RPI精确计算了绝热和非绝热模型系统的速率常数，其耦合强度变化了三个数量级。并且我们从数字上发现这些解决方案是真实的实例，具有一阶鞍点所期望的单个不稳定模式。最后，我们使用MF-RPI精确计算了绝热和非绝热模型系统的速率常数，其耦合强度变化了三个数量级。