量子经典分子动力学时间分裂方案的可观测误差界
Observable Error Bounds of the Time-splitting Scheme for Quantum-Classical Molecular Dynamics
摘要：
量子经典分子动力学作为全量子薛定谔方程的部分经典极限，是量子分子动力学的一个广泛应用的框架。基本方程本质上是非线性的，包含量子部分（代表电子）和经典部分（代表原子核）。波函数的精确模拟通常需要与重新标度的普朗克常数$h$相当的时间步长，从而导致当$h\ll 1$。基于半经典分析，证明了所提出的时间分裂方案的Schwartz观测量的可加性观测误差界，该误差界随着$h$的减小而减小.此外，我们建立了一个统一的$h$可观测误差界，它允许一个$\mathcal{O}(1)$时间步长，以准确地捕捉物理可观测的大小无关的$h$。数值结果验证了我们的估计。
英文摘要：
Quantum-classical molecular dynamics, as a partial classical limit of the full quantum Schrödinger equation, is a widely used framework for quantum molecular dynamics. The underlying equations are nonlinear in nature, containing a quantum part (represents the electrons) and a classical part (stands for the nuclei). An accurate simulation of the wave function typically requires a time step comparable to the rescaled Planck constant $h$, resulting in a formidable cost when $h\ll 1$. We prove an additive observable error bound of Schwartz observables for the proposed time-splitting schemes based on semiclassical analysis, which decreases as $h$ becomes smaller. Furthermore, we establish a uniform-in-$h$ observable error bound, which allows an $\mathcal{O}(1)$ time step to accurately capture the physical observable regardless of the size of $h$. Numerical results verify our estimates.
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