Efficient importance sampling in semiclassical initial value representation calculations for time correlation functions

Regular Article


The semiclassical (SC) theory based on an initial value representation (IVR) methodology provides a practical way to describe quantum effects in complex molecular systems. The efficiency of SC–IVR calculations for time correlation functions depends heavily on how to perform the Monte Carlo sampling of initial conditions. Here, we compare a variety of possibilities of sampling initial conditions in the SC calculations by choosing the sampling function to be either time-dependent or time-independent (TI). The implementation of these importance sampling protocols to two benchmark system-bath models demonstrates its advantages over the standard sampling method. In particular, the recently developed TI importance sampling which incorporates path correlation in the bath degrees of freedom shows a great potential in describing many-body quantum dynamics efficiently and accurately.


Semiclassical Time correlation function Importance sampling Quantum dynamics Molecular dynamics 



G.T. is grateful to Bill Miller for his critical reading of the original manuscript and helpful suggestions. This work was supported by Peking University Shenzhen Graduate School, and the Shenzhen Strategic Emerging Industry Development Funds (Grant No. JCYJ20120829170028565 and ZDSY20130331145131323).We also acknowledge a generous allocation of supercomputing time from the National Supercomputing Center in Shenzhen (Shenzhen Cloud Computing Center).


  1. 1.
    Clary DC (2008) Quantum dynamics of chemical reactions. Science 321:789–791CrossRefGoogle Scholar
  2. 2.
    Xiao C, Xu X, Liu S, Wang T, Dong W, Yang T, Sun Z, Dai D, Xu X, Zhang DH, Yang X (2011) Experimental and theoretical differential cross sections for a four-atom reaction: HD + OH → H2O + D. Science 333:440–442CrossRefGoogle Scholar
  3. 3.
    Kosloff R (1988) Time-dependent quantum-mechanical methods for molecular dynamics. J Phys Chem 92:2087–2100CrossRefGoogle Scholar
  4. 4.
    Beck MH, Jackle A, Worth GA, Meyer HD (2000) The multiconfiguration time-dependent Hartree (MCTDH) method: a highly efficient algorithm for propagating wavepackets. Phys Rep 324:1–105CrossRefGoogle Scholar
  5. 5.
    Wang H, Thoss M (2003) Multilayer formulation of the multiconfiguration time-dependent Hartree theory. J Chem Phys 119:1289–1299CrossRefGoogle Scholar
  6. 6.
    Meyer HD, Worth GA (2003) Quantum molecular dynamics: propagating wavepackets and density operators using the multiconfiguration time-dependent Hartree method. Theor Chem Acc 109:251–267CrossRefGoogle Scholar
  7. 7.
    Lopez-Lopez S, Martinazzo R, Nest M (2011) Benchmark calculations for dissipative dynamics of a system coupled to an anharmonic bath with the multi-configuration time-dependent Hartree method. J Chem Phys 134:094102CrossRefGoogle Scholar
  8. 8.
    Makri N (1999) Time-dependent quantum methods for large systems. Annu Rev Phys Chem 50:167–191CrossRefGoogle Scholar
  9. 9.
    Topaler M, Makri N (1994) Quantum rates for a double well coupled to a dissipative bath: accurate path integral results and comparison with approximation theories. J Chem Phys 101:7500–7519CrossRefGoogle Scholar
  10. 10.
    Billing GD (1975) On the applicability of the classical trajectory equations in inelastic scattering theory. Chem Phys Lett 30:391–393CrossRefGoogle Scholar
  11. 11.
    Billing GD (1993) Quantum corrections to the classical path theory. J Chem Phys 99:5849–5857CrossRefGoogle Scholar
  12. 12.
    Tully JC (1990) Molecular dynamics with electronic transitions. J Chem Phys 93:1061–1071CrossRefGoogle Scholar
  13. 13.
    Webster F, Schnitker J, Friedrichs MS, Friesner RA, Rossky PJ (1991) Solvation dynamics of the hydrated electron: a nonadiabatic quantum simulation. Phys Rev Lett 66:3172–3175CrossRefGoogle Scholar
  14. 14.
    Xiao L, Coker DF (1995) Methods for molecular dynamics with nonadiabatic transitions. J Chem Phys 102:496–510CrossRefGoogle Scholar
  15. 15.
    Antoniou D, Gelman D, Schwartz SD (2007) New mixed quantum/semiclassical propagation method. J Chem Phys 126:184107CrossRefGoogle Scholar
  16. 16.
    Chapman CT, Cina JA (2007) Semiclassical treatments for small-molecule dynamics in low-temperature crystals using fixed and adiabatic vibrational bases. J Chem Phys 127:114502CrossRefGoogle Scholar
  17. 17.
    Miller WH (1998) Spiers memorial lecture: quantum and semiclassical theory of chemical reaction rates. Faraday Discuss 110:1–21CrossRefGoogle Scholar
  18. 18.
    Miller WH (2001) The semiclassical initial value representation: a potentially practical way for adding quantum effects to classical molecular dynamics simulations. J Phys Chem A 105:2942–2955CrossRefGoogle Scholar
  19. 19.
    Miller WH (2009) Electronically nonadiabatic dynamics via semiclassical initial value methods. J Phys Chem A 113:1405–1415CrossRefGoogle Scholar
  20. 20.
    Miller WH (2012) Perspective: quantum or classical coherence? J Chem Phys 136:210901CrossRefGoogle Scholar
  21. 21.
    Tannor DJ, Garaschuk S (2000) Semiclassical calculation of chemical reaction dynamics via wave packet correlation functions. Annu Rev Phys Chem 51:553–600CrossRefGoogle Scholar
  22. 22.
    Thoss M, Wang H (2004) Semiclassical description of molecular dynamics based on initial-value representation methods. Annu Rev Phys Chem 55:299–302CrossRefGoogle Scholar
  23. 23.
    Kay KG (2005) Semiclassical initial value treatments of atoms and molecules. Annu Rev Phys Chem 56:255–280CrossRefGoogle Scholar
  24. 24.
    Kay KG (1994) Semiclassical propagation for multidimensional systems by an initial value method. J Chem Phys 101:2250–2260CrossRefGoogle Scholar
  25. 25.
    Makri N, Miller WH (1987) Time-dependent self consistent field (TDSCF) approximation for a reaction coordinate coupled to a harmonic bath: single and multi-configuration treatments. J Chem Phys 87:5781–5787CrossRefGoogle Scholar
  26. 26.
    Walton AR, Manolopoulos DE (1996) A new semiclassical initial-value method for Franck–Condon spectra. Mol Phys 87:961–978CrossRefGoogle Scholar
  27. 27.
    Wang H, Manolopoulos DE, Miller WH (2001) Generalized Filinov transformation of the semiclassical initial value representation. J Chem Phys 115:6317–6326CrossRefGoogle Scholar
  28. 28.
    Kaledin AL, Miller WH (2003) Time averaging the semiclassical initial value representation for the calculation of vibrational energy levels. J Chem Phys 118:7174–7182CrossRefGoogle Scholar
  29. 29.
    Ceotto M, Atahan S, Tantardini GF, Aspuru-Guzik A (2009) Multiple coherent states for first-principles semiclassical initial value representation molecular dynamics. J Chem Phys 130:234113CrossRefGoogle Scholar
  30. 30.
    Makri N, Thompson K (1998) Semiclassical influence functionals for quantum systems in anharmonic environments. Chem Phys Lett 291:101–109CrossRefGoogle Scholar
  31. 31.
    Sun X, Miller WH (1999) Forward–backward initial value representation for semiclassical time correlation functions. J Chem Phys 110:6635–6644CrossRefGoogle Scholar
  32. 32.
    Wang H, Thoss M, Sorge KL, Gelabert R, Giménez X, Miller WH (2001) Semiclassical description of quantum coherence effects and their quenching: a forward–backward initial value representation study. J Chem Phys 114:2562–2571CrossRefGoogle Scholar
  33. 33.
    Wang H, Sun X, Miller WH (1998) Semiclassical approximations for the calculation of thermal rate constants for chemical reactions in complex molecular systems. J Chem Phys 108:9726–9736CrossRefGoogle Scholar
  34. 34.
    Poulsen JA, Nyman G, Rossky PJ (2003) Practical evaluation of condensed phase quantum correlation functions: a Feynman–Kleinert variational linearized path integral method. J Chem Phys 119:12179–12193CrossRefGoogle Scholar
  35. 35.
    Shi Q, Geva E (2003) A relationship between semiclassical and centroid correlation functions. J Chem Phys 118:8173–8184CrossRefGoogle Scholar
  36. 36.
    Tao G, Miller WH (2011) Time-dependent importance sampling in semiclassical initial value representation calculations for time correlation functions. J Chem Phys 135:024104CrossRefGoogle Scholar
  37. 37.
    Tao G, Miller WH (2012) Time-dependent importance sampling in semiclassical initial value representation calculations for time correlation functions. II. A simplified implementation. J Chem Phys 137:124105CrossRefGoogle Scholar
  38. 38.
    Tao G, Miller WH (2013) Time-dependent importance sampling in semiclassical initial value representation calculations for time correlation functions. III. A state-resolved implementation to electronically non-adiabatic dynamics. Mol Phys 111:1987–1993CrossRefGoogle Scholar
  39. 39.
    Pan F, Tao G (2013) Importance sampling including path correlation in semiclassical initial value representation calculations for time correlation functions. J Chem Phys 138:091101CrossRefGoogle Scholar
  40. 40.
    Tao G (2013) Electronically non-adiabatic dynamics in complex molecular systems: an efficient and accurate semiclassical solution. J Phys Chem A 117:5821–5825CrossRefGoogle Scholar
  41. 41.
    Burant JC, Batista VS (2002) Real time path integrals using the Herman–Kluk propagator. J Chem Phys 116:2748–2756CrossRefGoogle Scholar
  42. 42.
    Yamamoto T, Wang H, Miller WH (2002) Combining semiclassical time evolution and quantum Boltzmann operator to evaluate reactive flux correlation function for thermal rate constants of complex systems. J Chem Phys 116:7335–7349CrossRefGoogle Scholar
  43. 43.
    Miller WH (1974) Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants. J Chem Phys 61:1823–1834CrossRefGoogle Scholar
  44. 44.
    Miller WH, Schwartz SD, Thromp JW (1983) Quantum mechanical rate constants for bimolecular reactions. J Chem Phys 79:4889–4898CrossRefGoogle Scholar
  45. 45.
    Berne BJ, Harp GD (1970) On the calculation of time correlation functions. Adv Chem Phys 17:63–227Google Scholar
  46. 46.
    Herman MF, Kluk E (1984) Semiclassical justification for the use of non-spreading wave packets in dynamics calculations. Chem Phys 91:27–34CrossRefGoogle Scholar
  47. 47.
    Kluk E, Herman MF, Davis HL (1986) Comparison of the propagation of semiclassical frozen Gaussian wave functions with quantum propagation for a highly excited anharmonic oscillator. J Chem Phys 84:326–334CrossRefGoogle Scholar
  48. 48.
    Shao J, Makri N (1999) Forward–backward semiclassical dynamics without prefactors. J Phys Chem A 103:7753–7756CrossRefGoogle Scholar
  49. 49.
    Fang JY, Martens CC (1996) An effective Hamiltonian-based method for mixed quantum-classical dynamics on coupled electronic surfaces. J Chem Phys 104:3684–3691CrossRefGoogle Scholar
  50. 50.
    Tao G, Miller WH (2009) Semiclassical description of vibrational quantum coherence in a three dimensional I2Arn (n ≤ 6) cluster: a forward-backward initial value representation implementation. J Chem Phys 130:184108CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Shenzhen Key Laboratory of New Energy Materials by DesignPeking UniversityShenzhenChina
  2. 2.School of Advanced Materials, Shenzhen Graduate SchoolPeking UniversityShenzhenChina

Personalised recommendations