Observable Error Bounds of the Time-splitting Scheme for Quantum-Classical Molecular Dynamics

• URL: http://arxiv.org/abs/2108.08245v2
• Date: Thu, 13 Oct 2022 17:36:11 GMT
• Title: Observable Error Bounds of the Time-splitting Scheme for Quantum-Classical Molecular Dynamics
• Authors: Di Fang and Albert Tres
• Abstract summary: We prove an additive observable error bound of Schwartz observables for the proposed time-splitting schemes. We establish a uniform-in-$h$ observable error bound, which allows an $mathcalO(1)$ time step to accurately capture the physical observable.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Quantum-classical molecular dynamics, as a partial classical limit of the full quantum Schr\"odinger 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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