Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A.I. Arnold. Mathematical methods of classical mechanics. Springer-Verlag, 1978.
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.
H.J.C. Berendsen and J. Mavri. Quantum simulation of reaction dynamics by density matrix evolution. J. Phys. Chem., 97: 13464–13468, 1993.
J.J. Biesiadecki and R.D. Skeel. Dangers of multiple-time-step methods. J. Comput. Phys., 109: 318–328, 1993.
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.
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.
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.
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.
S.K. Gray and D.E. Manolopoulos. Symplectic integrators tailored to the timedependent Schrödinger equation. J. Chem. Phys., 104: 7099–7112, 1996.
M. Hochbruck and Ch. Lubich. A bunch of time integrators for quantum/classical molecular dynamics, this volume.
E. Hairer and Ch. Lubich. The life-span of backward error analysis for numerical integrators. Numer. Math., 76: 441–462, 1997.
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.
P. Nettesheim and Ch. Schütte. Numerical integrators for quantum-classical molecular dynamics, this volume.
S. Reich. Backward error analysis for numerical integrators. SIAM J. Numer. Anal., to appear, 1999.
S. Reich. Preservation of adiabatic invariants under symplectic discretization. Applied Numerical Mathematics, to appear, 1998.
U. Schmitt and J. Brinkmann. Discrete time-reversible propagation scheme for mixed quantum classical dynamics. Chem. Phys., 208: 45–56, 1996.
U. Schmitt. Gemischt klassisch-quantenmechanische Molekulardynamik im Liouville-Formalismus. Ph.D. thesis (in german), Darmstadt, 1997.
J.M. Sanz-Serna and M.P. Calvo. Numerical Hamiltonian Systems. Chapman and Hall, London, 1994.
M. Tuckerman, B.J. Berne, and G. Martyna. Reversible Multiple Time Scale Molecular Dynamics. J. Chem. Phys., 97: 1990–2001, 1992.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Nettesheim, P., Reich, S. (1999). Symplectic Multiple-Time-Stepping Integrators for Quantum-Classical Molecular Dynamics. In: Deuflhard, P., Hermans, J., Leimkuhler, B., Mark, A.E., Reich, S., Skeel, R.D. (eds) Computational Molecular Dynamics: Challenges, Methods, Ideas. Lecture Notes in Computational Science and Engineering, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58360-5_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-58360-5_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63242-9
Online ISBN: 978-3-642-58360-5
eBook Packages: Springer Book Archive