Numerische Mathematik

, Volume 137, Issue 1, pp 119–157 | Cite as

Discretising the Herman–Kluk propagator

  • Caroline Lasser
  • David Sattlegger


The Herman–Kluk propagator is a well-known semi-classical approximation of the unitary evolution operator in quantum molecular dynamics. In this paper we formulate the Herman–Kluk propagator as a phase space integral and discretise it by Monte Carlo and quasi-Monte Carlo quadrature. Then, we investigate the accuracy of a symplectic time discretisation by combining backward error analysis with Fourier integral operator calculus. Numerical experiments for two- and six-dimensional model systems support our theoretical results.


Herman–Kluk propagator Semi-classical approximation Mesh-less discretisation Symplectic methods 

Mathematics Subject Classification

81Q20 65D30 65Z05 65P10 



The authors gratefully acknowledge the support by the DFG through the International Research Training Group IGDK 1754 “Optimisation and Numerical Analysis for Partial Differential Equations with Non-smooth Structures”.


  1. 1.
    Aistleitner, C., Dick, J.: Functions of bounded variation, signed measures, and a general Koksma–Hlawka inequality. Acta Arith. 167(2), 143–171 (2015). doi: 10.4064/aa167-2-4 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Faou, E., Gradinaru, V., Lubich, C.: Computing semiclassical quantum dynamics with Hagedorn wavepackets. SIAM J. Sci. Comput. 31(4), 3027–3041 (2009). doi: 10.1137/080729724 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Faou, E., Lubich, C.: A poisson integrator for gaussian wavepacket dynamics. Comput. Vis. Sci. 9(2), 45–55 (2006). doi: 10.1007/s00791-006-0019-8 MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gradinaru, V., Hagedorn, G.A.: Convergence of a semiclassical wavepacket based time-splitting for the Schrödinger equation. Numer. Math. 126(1), 53–73 (2014). doi: 10.1007/s00211-013-0560-6 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hagedorn, G.A.: Semiclassical quantum mechanics. Comm. Math. Phys. 71, 77–93 (1980). doi: 10.1007/BF01230088 MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hagedorn, G.A.: Raising and lowering operators for semiclassical wave packets. Ann. Phys. 269(1), 77–104 (1998). doi: 10.1006/aphy.1998.5843 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, 2nd edn. No. 31 in Springer Series in Computational Mathematics. Springer, Berlin (2006). doi: 10.1007/3-540-30666-8
  8. 8.
    Harabati, C., Rost, J., Grossmann, F.: Long-time and unitary properties of semiclassical initial value representations. J. Chem. Phys. 120(1), 26–30 (2004). doi: 10.1063/1.1630033 CrossRefGoogle Scholar
  9. 9.
    Heller, E.J.: Time dependent variational approach to semiclassical dynamics. J. Chem. Phys. 64(1), 63–73 (1976). doi: 10.1063/1.431911 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Heller, E.J.: Frozen Gaussians: a very simple semiclassical approximation. J. Chem. Phys. 75(6), 2923–2931 (1981). doi: 10.1063/1.442382 MathSciNetCrossRefGoogle Scholar
  11. 11.
    Herman, M.F., Kluk, E.: A semiclassical justification for the use of non-spreading wavepackets in dynamics calculations. Chem. Phys. 91(1), 27–34 (1984). doi: 10.1016/0301-0104(84)80039-7 CrossRefGoogle Scholar
  12. 12.
    Kahan, W., Li, R.C.: Composition constants for raising the orders of unconventional schemes for ordinary differential equations. Math. Comput. 66(219), 1089–1099 (1997). doi: 10.1090/S0025-5718-97-00873-9 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lasser, C., Röblitz, S.: Computing expectation values for molecular quantum dynamics. SIAM J. Sci. Comput. 32(3), 1465–1483 (2010). doi: 10.1137/090770461 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lasser, C., Troppmann, S.: Hagedorn wavepackets in time-frequency and phase space. J. Fourier Anal. Appl. 20(4), 679–714 (2014). doi: 10.1007/s00041-014-9330-9 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Littlejohn, R.G.: The semiclassical evolution of wave packets. Phys. Rep. 138(4–5), 193–291 (1986). doi: 10.1016/0370-1573(86)90103-1 MathSciNetCrossRefGoogle Scholar
  16. 16.
    Liu, H., Runborg, O., Tanushev, N.M.: Error estimates for Gaussian beam superpositions. Math. Comput. 82(282), 919–952 (2013). doi: 10.1090/S0025-5718-2012-02656-1 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Martinez, A.: An Introduction to Semiclassical and Microlocal Analysis. Universitext. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  18. 18.
    Miller, W.H.: Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants. J. Chem. Phys. 61(5), 1823–1834 (1974). doi: 10.1063/1.1682181 CrossRefGoogle Scholar
  19. 19.
    Swart, T., Rousse, V.: A mathematical justification for the Herman–Kluk propagator. Comm. Math. Phys. 286(2), 725–750 (2009). doi: 10.1007/s00220-008-0681-4 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Thoss, M., Wang, H.: Semiclassical description of molecular dynamics based on initial-value representation methods. Annu. Rev. Phys. Chem. 55(1), 299–332 (2004). doi: 10.1146/annurev.physchem.55.091602.094429 CrossRefGoogle Scholar
  21. 21.
    Zhen, C.: Optimal error estimates for first-order Gaussian beam approximations to the Schrödinger equation. SIAM J. Numer. Anal. 52(6), 2905–2930 (2014). doi: 10.1137/130935720 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Technische Universität MünchenMunichGermany

Personalised recommendations