# An Efficient Time-splitting Method for the Ehrenfest Dynamics

## TL;DR

This paper introduces a stable and efficient time-splitting numerical method for Ehrenfest dynamics, enabling accurate quantum-classical simulations with larger time steps independent of the semiclassical parameter.

## Contribution

A novel time-splitting scheme for Ehrenfest dynamics that is stable, preserves the semiclassical limit, and allows larger time steps for efficient simulations.

## Key findings

- Scheme is stable uniformly with respect to the semiclassical parameter.
- Method preserves the discrete semiclassical limit.
- Numerical examples confirm the effectiveness of the meshing strategy.

## Abstract

The Ehrenfest dynamics, representing a quantum-classical mean-field type coupling, is a widely used approximation in quantum molecular dynamics. In this paper, we propose a time-splitting method for an Ehrenfest dynamics, in the form of a nonlinearly coupled Schr\"odinger-Liouville system. We prove that our splitting scheme is stable uniformly with respect to the semiclassical parameter, and, moreover, that it preserves a discrete semiclassical limit. Thus one can accurately compute physical observables using time steps induced only by the classical Liouville equation, i.e., independent of the small semiclassical parameter - in addition to classical mesh sizes for the Liouville equation. Numerical examples illustrate the validity of our meshing strategy.