Department of Chemistry, New York University, New York, New York 10003, USA.
J Chem Phys. 2018 Mar 14;148(10):102340. doi: 10.1063/1.5005543.
We introduce a scheme for approximating quantum time correlation functions numerically within the Feynman path integral formulation. Starting with the symmetrized version of the correlation function expressed as a discretized path integral, we introduce a change of integration variables often used in the derivation of trajectory-based semiclassical methods. In particular, we transform to sum and difference variables between forward and backward complex-time propagation paths. Once the transformation is performed, the potential energy is expanded in powers of the difference variables, which allows us to perform the integrals over these variables analytically. The manner in which this procedure is carried out results in an open-chain path integral (in the remaining sum variables) with a modified potential that is evaluated using imaginary-time path-integral sampling rather than requiring the generation of a large ensemble of trajectories. Consequently, any number of path integral sampling schemes can be employed to compute the remaining path integral, including Monte Carlo, path-integral molecular dynamics, or enhanced path-integral molecular dynamics. We believe that this approach constitutes a different perspective in semiclassical-type approximations to quantum time correlation functions. Importantly, we argue that our approximation can be systematically improved within a cumulant expansion formalism. We test this approximation on a set of one-dimensional problems that are commonly used to benchmark approximate quantum dynamical schemes. We show that the method is at least as accurate as the popular ring-polymer molecular dynamics technique and linearized semiclassical initial value representation for correlation functions of linear operators in most of these examples and improves the accuracy of correlation functions of nonlinear operators.
我们提出了一种在费曼路径积分公式中数值逼近量子时间相关函数的方案。从表示为离散路径积分的相关函数的对称化版本开始，我们引入了在基于轨迹的半经典方法推导中经常使用的积分变量变换。特别是，我们将正向和反向复时间传播路径之间的求和变量和差变量转换。一旦进行了变换，势能就可以展开为差变量的幂次，这允许我们对这些变量进行解析积分。执行此过程的方式导致具有修改后的势能的开链路径积分（在剩余的求和变量中），该势能是使用虚时间路径积分采样而不是需要生成大量轨迹的集合来评估的。因此，可以采用任何数量的路径积分采样方案来计算剩余的路径积分，包括蒙特卡罗、路径积分分子动力学或增强的路径积分分子动力学。我们认为这种方法构成了量子时间相关函数的半经典型近似的不同视角。重要的是，我们认为我们的近似可以在累积展开形式主义中系统地改进。我们在一组常用于基准近似量子动力学方案的一维问题上测试了这种近似。我们表明，该方法在大多数情况下至少与流行的环聚合物分子动力学技术和线性化半经典初始值表示一样准确，并且可以提高非线性算子相关函数的准确性。
