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Exponential Integrators for Quantum-Classical Molecular Dynamics

BIT Numerical Mathematics volume 39, pages 620–645 (1999)Cite this article

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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Authors and Affiliations

  1. Mathematisches Institut, Universität Düsseldorf, DE-40225, Düsseldorf, Germany

    Marlis Hochbruck

  2. Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, DE-72076, Tübingen, Germany, email: lubich@na.uni-tuebingen.de

    Christian Lubich

Authors
  1. Marlis Hochbruck
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  2. Christian Lubich
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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