Symplectic Multiple-Time-Stepping Integrators for Quantum-Classical Molecular Dynamics

  • Peter Nettesheim
  • Sebastian Reich
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 4)


The overall Hamiltonian structure of the Quantum-Classical Molecular Dynamics model makes — analogously to classical molecular dynamics - symplectic integration schemes the methods of choice for long-term simulations. This has already been demonstrated by the symplectic PICKABACK method [19]. However, this method requires a relatively small step-size due to the high-frequency quantum modes. Therefore, following related ideas from classical molecular dynamics, we investigate symplectic multiple-time-stepping methods and indicate various possibilities to overcome the step-size limitation of PICKABACK.


Classical Particle Classical Molecular Dynamic Symplectic Integrator Adiabatic Invariant Symplectic Method 
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  1. 1.
    A.I. Arnold. Mathematical methods of classical mechanics. Springer-Verlag, 1978.Google Scholar
  2. 2.
    G. Benettin and A. Giorgilli. On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms. J. Statist Phys., 74: 1117–1143, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    H.J.C. Berendsen and J. Mavri. Quantum simulation of reaction dynamics by density matrix evolution. J. Phys. Chem., 97: 13464–13468, 1993.CrossRefGoogle Scholar
  4. 4.
    J.J. Biesiadecki and R.D. Skeel. Dangers of multiple-time-step methods. J. Comput. Phys., 109: 318–328, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    F.A. Bornemann, P. Nettesheim, and Ch. Schütte. Quantum-classical molecular dynamics as an approximation for full quantum dynamics. J. Chem. Phys., 105(3): 1074–1083, 1996.CrossRefGoogle Scholar
  6. 6.
    F.A. Bornemann and Ch. Schütte. On the singular limit of the quantumclassical molecular dynamics model. Preprint SC 97–07, ZIB Berlin, 1997. Submitted to SIAM J. Appl. Math.Google Scholar
  7. 7.
    A. García-Vela, R.B. Gerber, and D.G. Imre. Mixed quantum wave packet/classical trajectory treatment of the photodissociation process ArHCl → Ar+H+Cl. J. Chem. Phys., 97: 7242–7250, 1992.CrossRefGoogle Scholar
  8. 8.
    R.B. Gerber, V. Buch, and M.A. Ratner. Time-dependent self-consistent field approximation for intramolecular energy transfer. J. Chem. Phys., 66: 3022–3030, 1982.CrossRefGoogle Scholar
  9. 9.
    S.K. Gray and D.E. Manolopoulos. Symplectic integrators tailored to the timedependent Schrödinger equation. J. Chem. Phys., 104: 7099–7112, 1996.CrossRefGoogle Scholar
  10. 10.
    M. Hochbruck and Ch. Lubich. A bunch of time integrators for quantum/classical molecular dynamics, this volume.Google Scholar
  11. 11.
    E. Hairer and Ch. Lubich. The life-span of backward error analysis for numerical integrators. Numer. Math., 76: 441–462, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    P. Nettesheim, F.A. Bornemann, B. Schmidt, and Ch. Schütte. An explicit and symplectic integrator for quantum-classical molecular dynamics. Chemical Physics Letters, 256: 581–588, 1996.CrossRefGoogle Scholar
  13. 13.
    P. Nettesheim and Ch. Schütte. Numerical integrators for quantum-classical molecular dynamics, this volume.Google Scholar
  14. 14.
    S. Reich. Backward error analysis for numerical integrators. SIAM J. Numer. Anal., to appear, 1999.Google Scholar
  15. 15.
    S. Reich. Preservation of adiabatic invariants under symplectic discretization. Applied Numerical Mathematics, to appear, 1998.Google Scholar
  16. 16.
    U. Schmitt and J. Brinkmann. Discrete time-reversible propagation scheme for mixed quantum classical dynamics. Chem. Phys., 208: 45–56, 1996.CrossRefGoogle Scholar
  17. 17.
    U. Schmitt. Gemischt klassisch-quantenmechanische Molekulardynamik im Liouville-Formalismus. Ph.D. thesis (in german), Darmstadt, 1997.Google Scholar
  18. 18.
    J.M. Sanz-Serna and M.P. Calvo. Numerical Hamiltonian Systems. Chapman and Hall, London, 1994.Google Scholar
  19. 19.
    M. Tuckerman, B.J. Berne, and G. Martyna. Reversible Multiple Time Scale Molecular Dynamics. J. Chem. Phys., 97: 1990–2001, 1992.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Peter Nettesheim
    • 1
  • Sebastian Reich
    • 1
  1. 1.Konrad-Zuse-ZentrumBerlinGermany

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