Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA.
Department of Mathematical Sciences Rutgers University Camden, Camden, New Jersey 08102, USA.
J Chem Phys. 2019 Sep 28;151(12):124103. doi: 10.1063/1.5120282.
Path-integral-based molecular dynamics (MD) simulations are widely used for the calculation of numerically exact quantum Boltzmann properties and approximate dynamical quantities. A nearly universal feature of MD numerical integration schemes for equations of motion based on imaginary-time path integrals is the use of harmonic normal modes for the exact evolution of the free ring-polymer positions and momenta. In this work, we demonstrate that this standard practice creates numerical artifacts. In the context of conservative (i.e., microcanonical) equations of motion, it leads to numerical instability. In the context of thermostated (i.e., canonical) equations of motion, it leads to nonergodicity of the sampling. These pathologies are generally proven to arise at integration time steps that depend only on the system temperature and the number of ring-polymer beads, and they are numerically demonstrated for the cases of conventional ring-polymer MD (RPMD) and thermostated RPMD (TRPMD). Furthermore, it is demonstrated that these numerical artifacts are removed via replacement of the exact free ring-polymer evolution with a second-order approximation based on the Cayley transform. The Cayley modification introduced here can immediately be employed with almost every existing integration scheme for path-integral-based MD-including path-integral MD (PIMD), RPMD, TRPMD, and centroid MD-providing strong symplectic stability and ergodicity to the numerical integration, at no penalty in terms of computational cost, algorithmic complexity, or accuracy of the overall MD time step. Furthermore, it is shown that the improved numerical stability of the Cayley modification allows for the use of larger MD time steps. We suspect that the Cayley modification will therefore find useful application in many future path-integral-based MD simulations.
基于路径积分的分子动力学 (MD) 模拟广泛用于计算数值精确的量子玻尔兹曼性质和近似动力学量。基于虚时间路径积分的运动方程的 MD 数值积分方案的一个几乎普遍的特征是使用调和正则模来精确演化自由环聚合物的位置和动量。在这项工作中，我们证明了这种标准做法会产生数值伪影。在保守（即微正则）运动方程的情况下，它会导致数值不稳定性。在恒温（即正则）运动方程的情况下，它会导致采样的非遍历性。这些病态通常被证明是在仅取决于系统温度和环聚合物珠数的积分时间步长下出现的，并且针对常规环聚合物 MD (RPMD) 和恒温 RPMD (TRPMD) 的情况进行了数值演示。此外，还证明通过用基于 Cayley 变换的二阶近似替换精确的自由环聚合物演化，可以消除这些数值伪影。这里引入的 Cayley 修正可以立即应用于几乎所有基于路径积分的 MD 的现有积分方案，包括路径积分 MD (PIMD)、RPMD、TRPMD 和质心 MD，为数值积分提供强的辛稳定性和遍历性，而不会对计算成本、算法复杂性或整体 MD 时间步长的准确性造成任何影响。此外，还表明 Cayley 修正的改进数值稳定性允许使用更大的 MD 时间步长。我们怀疑 Cayley 修正将在许多未来的基于路径积分的 MD 模拟中找到有用的应用。
