## Abstract

We study time integration methods for equations of mixed quantum-classical molecular dynamics in which Newtonian equations of motion and Schrödinger equations are nonlinearly coupled. Such systems exhibit different time scales in the classical and the quantum evolution, and the solutions are typically highly oscillatory. The numerical methods use the exponential of the quantum Hamiltonian whose product with a state vector is approximated using Lanczos' method. This allows time steps that are much larger than the inverse of the highest frequencies.

We describe various integration schemes and analyze their error behaviour, without assuming smoothness of the solution. As preparation and as a problem of independent interest, we study also integration methods for Schrödinger equations with time-dependent Hamiltonian.

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Hochbruck, M., Lubich, C. Exponential Integrators for Quantum-Classical Molecular Dynamics.
*BIT Numerical Mathematics* **39**, 620–645 (1999). https://doi.org/10.1023/A:1022335122807

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DOI: https://doi.org/10.1023/A:1022335122807

- Numerical integrator
- oscillatory solutions
- Schrödinger equation
- quantum-classical coupling
- error bounds
- stability