The European Physical Journal Special Topics

, Volume 224, Issue 12, pp 2305–2320 | Cite as

An introduction to the problem of bridging quantum and classical dynamics

  • S. Bonella
  • G. Ciccotti
Review A. Representation of Molecular Systems Across Scales
Part of the following topical collections:
  1. Discussion and Debate: Recurrent Problems in Scale Bridging Techniques in Molecular Simulation – What are the Current Options?


Simulating the exact quantum dynamics of realistic interacting systems is presently a task beyond reach but for the smallest of them, as the numerical cost for solving the time-dependent Schrödinger equation scales exponentially with the number of degrees of freedom. Mixed quantum-classical methods attempt to solve this problem by starting from a full quantum description of the system and subsequently partitioning the degrees of freedom in two subsets: the quantum subsystem and the bath. A classical limit is then taken for the bath while preserving, at least approximately, the quantum evolution of the subsystem. A key, as yet not fully resolved, theoretical question is how to do so by constructing a consistent description of the overall dynamics. An exhaustive review of this class of methods is beyond the scope of this paper and we shall limit ourselves to present, as an example, a specific approach, known as the LANDM-Map method. The method stems from an attempt at taking a rigorous limit for the classical degrees of freedom starting from a path integral formulation of the full quantum problem. The results that we discuss are not new, but our intent here is to present them as an introduction to the problem of mixed quantum classical dynamics. We shall also indicate a broad classification of the available approaches, their limitations, and some open questions in this field.


European Physical Journal Special Topic Molecular Simulation Time Correlation Function Nuclear Position Path Integral Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    P. Ehrenfest, Zeitung Physik 45, 455 (1927)CrossRefADSzbMATHGoogle Scholar
  2. 2.
    X. Li, J.C. Tully, H.B. Schlagel, M.J. Frisch, J. Chem. Phys. 123, 084106 (2005)CrossRefADSGoogle Scholar
  3. 3.
    J.C. Tully, J. Chem. Phys. 55, 562 (1971)CrossRefADSGoogle Scholar
  4. 4.
    J.C. Tully, J. Chem. Phys. 93, 1061 (1990)CrossRefADSGoogle Scholar
  5. 5.
    M. Barbatti, Wiley Interdisciplinary Reviews: Computational Molecular Science 1, 620 (2011)Google Scholar
  6. 6.
    E. Tapavicza, I. Tavernelli, U. Rothlisberger, Phys. Rev. Lett. 98, 023001 (2007)CrossRefADSGoogle Scholar
  7. 7.
    I. Tavernelli, E. Tapavicza, U. Rothlisberger, J. Chem. Phys. 130, 124107 (2009)CrossRefADSGoogle Scholar
  8. 8.
    I. Tavernelli, B.F.E. Curchod, U. Rothlisberger, Chem. Phys. 391, 101 (2011)CrossRefADSGoogle Scholar
  9. 9.
    R. Kapral, J. Phys.: Condens. Mat. 27, 1 (2015)Google Scholar
  10. 10.
    F. de Carvalho, M. Bouduban, B. Curchod, I. Tavernelli, Entropy 16, 62 (2014)MathSciNetCrossRefADSGoogle Scholar
  11. 11.
    B.J. Schwartz, E.R. Bittner, O.V. Prezhdo, P.J. Rossky, J. Chem. Phys. 104, 5942 (1996)CrossRefADSGoogle Scholar
  12. 12.
    G. Granucci, M. Persico, A. Zoccante, J. Chem. Phys. 133, 134111Google Scholar
  13. 13.
    J. Subotnik, W. Ouyang, B. Landry, J. Chem. Phys. 139, 214107 (2013)CrossRefADSGoogle Scholar
  14. 14.
    R. Kapral, G. Ciccotti, J. Chem. Phys. 110, 8919 (1999)CrossRefADSGoogle Scholar
  15. 15.
    E. Wigner, Phys. Rev. 40, 749 (1932)CrossRefADSGoogle Scholar
  16. 16.
    D.M. Kernan, G. Ciccotti, R. Kapral, J. Phys. Chem. B 112, 424 (2008)CrossRefGoogle Scholar
  17. 17.
    H. Kim, A. Nassimi, R. Kapral, J. Chem. Phys. 129, 084102 (2008)CrossRefADSGoogle Scholar
  18. 18.
    S. Nielsen, R. Kapral, G. Ciccotti, J. Chem. Phys. 115, 5805 (2001)CrossRefADSGoogle Scholar
  19. 19.
    S. Bonella, D.F. Coker, J. Chem. Phys. 122, 194102 (2005)CrossRefADSGoogle Scholar
  20. 20.
    S. Bonella, D. Montemayor, D.F. Coker, Proc. Natl Acad. Sci. USA 102, 6715 (2005)CrossRefADSGoogle Scholar
  21. 21.
    S. Bonella, R. Kapral, G. Ciccotti, Chem. Phys. Lett. 484, 399 (2010)CrossRefADSGoogle Scholar
  22. 22.
    D. Coker, S. Bonella, Computer Simulations in Condensed Matter: From Materials to Chemical Biology, Vol. 1, Lecture Notes in Physics (Springer, 2006), p. 553Google Scholar
  23. 23.
    F. Strocchi, Rev. Mod. Phys. 38, 36 (1966)MathSciNetCrossRefADSzbMATHGoogle Scholar
  24. 24.
    G. Stock, M. Thoss, Phys. Rev. Lett. 78, 578 (1997)MathSciNetCrossRefADSzbMATHGoogle Scholar
  25. 25.
    G. Stock, M. Thoss, Phys. Rev. A 59, 64 (1999)MathSciNetCrossRefADSGoogle Scholar
  26. 26.
    W.H. Miller, C.W. McCurdy, J. Chem. Phys. 69, 5163 (1978)CrossRefADSGoogle Scholar
  27. 27.
    C.W. McCurdy, H.D. Meyer, W.H. Miller, J. Chem. Phys. 70, 3177 (1979)CrossRefADSGoogle Scholar
  28. 28.
    R.P. Feynman, Rev. Mod. Phys. 20, 367 (1948)MathSciNetCrossRefADSGoogle Scholar
  29. 29.
    H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, and Financial Markets (Oxford University Press, Oxford, 2004)Google Scholar
  30. 30.
    L.S. Schulman, Techniques and Applications of Path Integration (Dover Books on Physics) (Dover Publications, 2005)Google Scholar
  31. 31.
    Q. Shi, E. Geva, J. Phys. Chem. A 107, 9059 (2003)CrossRefGoogle Scholar
  32. 32.
    J.A. Poulsen, G. Nyman, P.J. Rossky, J. Chem. Phys. 119, 12179 (2003)CrossRefADSGoogle Scholar
  33. 33.
    J. Liu, W.H. Miller, J. Chem. Phys. 131, 074113 (2009)CrossRefADSGoogle Scholar
  34. 34.
    J. Beutier, D. Borgis, R. Vuilleumier, S. Bonella, J. Chem. Phys. 141, 084102 (2014)CrossRefADSGoogle Scholar
  35. 35.
    V.S. Filinov, Nucl. Phys. B 271, 717 (1986)CrossRefADSGoogle Scholar
  36. 36.
    N. Makri, W.H. Miller, Chem. Phys. Lett. 139, 10 (1987)CrossRefADSGoogle Scholar
  37. 37.
    M.S. Causo, G. Ciccotti, S. Bonella, R. Vuilleumier, J. Phys. Chem. B 110, 3638 (2006)CrossRefGoogle Scholar
  38. 38.
    M. Monteferrante, S. Bonella, G. Ciccotti, Molec. Phys. 109, 3015 (2011)CrossRefADSGoogle Scholar
  39. 39.
    D. Mac Kernan, G. Ciccotti, R. Kapral, J. Chem. Phys. 116, 2346 (2002)CrossRefADSGoogle Scholar
  40. 40.
    C.H. Mak, D. Chandler, Phys. Rev. A 44, 2352 (1991)CrossRefADSGoogle Scholar
  41. 41.
    M. Topaler, N. Makri, J. Chem. Phys. 101, 7500 (1994)CrossRefADSGoogle Scholar
  42. 42.
    R. Egger, C.H. Mak, Phys. Rev. B 50, 15210 (1994)CrossRefADSGoogle Scholar
  43. 43.
    K. Thompson, N. Makri, Chem. Phys. Lett. 291, 101 (1998)CrossRefADSGoogle Scholar
  44. 44.
    X. Sun, H.B. Wang, W.H. Miller, J. Chem. Phys. 109, 7064 (1998)CrossRefADSGoogle Scholar
  45. 45.
    K. Thompson, N. Makri, J. Chem. Phys. 110, 1343 (1999)CrossRefADSGoogle Scholar
  46. 46.
    A. Golosov, D.R. Reichman, J. Chem. Phys. 114, 1065 (2001)CrossRefADSGoogle Scholar
  47. 47.
    E.R. Dunkel, S. Bonella, D.F. Coker, J. Chem. Phys. 129, 114106 (2008)CrossRefADSGoogle Scholar
  48. 48.
    P. Huo, D.F. Coker, J. Chem. Phys. 133, 184108 (2010)CrossRefADSGoogle Scholar
  49. 49.
    P. Huo, T.F. Miller, D.F. Coker, J. Chem. Phys. 139, 151103 (2013)CrossRefADSGoogle Scholar
  50. 50.
    S. Bonella, M. Monteferrante, C. Pierleoni, G. Ciccotti, J. Chem. Phys. 133, 164105 (2010)CrossRefADSGoogle Scholar
  51. 51.
    J.J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994)Google Scholar

Copyright information

© EDP Sciences and Springer 2015

Authors and Affiliations

  1. 1.CECAM Centre Européen de Calcul Atomique et MoléculaireÉcole Polytechnique Fédérale de Lausanne, BatochimeLausanneSuisse
  2. 2.Dipartimento di FisicaUniversità di Roma “La Sapienza”RomaItalia
  3. 3.School of PhysicsUniversity College of Dublin UCDDublin 4Ireland

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