Department of Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.
J Chem Phys. 2018 May 21;148(19):194110. doi: 10.1063/1.5025058.
We introduce a new path integral Monte Carlo method for investigating nonadiabatic systems in thermal equilibrium and demonstrate an approach to reducing stochastic error. We derive a general path integral expression for the partition function in a product basis of continuous nuclear and discrete electronic degrees of freedom without the use of any mapping schemes. We separate our Hamiltonian into a harmonic portion and a coupling portion; the partition function can then be calculated as the product of a Monte Carlo estimator (of the coupling contribution to the partition function) and a normalization factor (that is evaluated analytically). A Gaussian mixture model is used to evaluate the Monte Carlo estimator in a computationally efficient manner. Using two model systems, we demonstrate our approach to reduce the stochastic error associated with the Monte Carlo estimator. We show that the selection of the harmonic oscillators comprising the sampling distribution directly affects the efficiency of the method. Our results demonstrate that our path integral Monte Carlo method's deviation from exact Trotter calculations is dominated by the choice of the sampling distribution. By improving the sampling distribution, we can drastically reduce the stochastic error leading to lower computational cost.
我们引入了一种新的路径积分蒙特卡罗方法，用于研究热平衡中的非绝热系统，并展示了一种降低随机误差的方法。我们在连续核和离散电子自由度的乘积基中推导出了一个没有使用任何映射方案的一般路径积分表示式来计算配分函数。我们将哈密顿量分为谐波部分和耦合部分；然后可以将配分函数计算为蒙特卡罗估计量（对配分函数的耦合贡献）和归一化因子（通过解析评估）的乘积。我们使用高斯混合模型以计算效率的方式来评估蒙特卡罗估计量。通过两个模型系统，我们展示了我们的方法来降低与蒙特卡罗估计量相关的随机误差。我们表明，构成采样分布的谐振子的选择直接影响方法的效率。我们的结果表明，我们的路径积分蒙特卡罗方法与精确 Trotter 计算的偏差主要取决于采样分布的选择。通过改进采样分布，我们可以大大降低随机误差，从而降低计算成本。
