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

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 hll 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.