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

TL;DR

This paper establishes error bounds for time-splitting schemes in quantum-classical molecular dynamics, enabling accurate simulations with larger time steps independent of the semiclassical parameter h.

Contribution

It provides the first uniform-in-h observable error bounds for time-splitting schemes in quantum-classical molecular dynamics, improving computational efficiency.

Findings

Error bounds decrease as h becomes smaller

Uniform-in-h error bounds allow larger time steps

Numerical results confirm theoretical estimates

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.