Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Ehrenfest, Zeitung Physik 45, 455 (1927)
X. Li, J.C. Tully, H.B. Schlagel, M.J. Frisch, J. Chem. Phys. 123, 084106 (2005)
J.C. Tully, J. Chem. Phys. 55, 562 (1971)
J.C. Tully, J. Chem. Phys. 93, 1061 (1990)
M. Barbatti, Wiley Interdisciplinary Reviews: Computational Molecular Science 1, 620 (2011)
E. Tapavicza, I. Tavernelli, U. Rothlisberger, Phys. Rev. Lett. 98, 023001 (2007)
I. Tavernelli, E. Tapavicza, U. Rothlisberger, J. Chem. Phys. 130, 124107 (2009)
I. Tavernelli, B.F.E. Curchod, U. Rothlisberger, Chem. Phys. 391, 101 (2011)
R. Kapral, J. Phys.: Condens. Mat. 27, 1 (2015)
F. de Carvalho, M. Bouduban, B. Curchod, I. Tavernelli, Entropy 16, 62 (2014)
B.J. Schwartz, E.R. Bittner, O.V. Prezhdo, P.J. Rossky, J. Chem. Phys. 104, 5942 (1996)
G. Granucci, M. Persico, A. Zoccante, J. Chem. Phys. 133, 134111
J. Subotnik, W. Ouyang, B. Landry, J. Chem. Phys. 139, 214107 (2013)
R. Kapral, G. Ciccotti, J. Chem. Phys. 110, 8919 (1999)
E. Wigner, Phys. Rev. 40, 749 (1932)
D.M. Kernan, G. Ciccotti, R. Kapral, J. Phys. Chem. B 112, 424 (2008)
H. Kim, A. Nassimi, R. Kapral, J. Chem. Phys. 129, 084102 (2008)
S. Nielsen, R. Kapral, G. Ciccotti, J. Chem. Phys. 115, 5805 (2001)
S. Bonella, D.F. Coker, J. Chem. Phys. 122, 194102 (2005)
S. Bonella, D. Montemayor, D.F. Coker, Proc. Natl Acad. Sci. USA 102, 6715 (2005)
S. Bonella, R. Kapral, G. Ciccotti, Chem. Phys. Lett. 484, 399 (2010)
D. Coker, S. Bonella, Computer Simulations in Condensed Matter: From Materials to Chemical Biology, Vol. 1, Lecture Notes in Physics (Springer, 2006), p. 553
F. Strocchi, Rev. Mod. Phys. 38, 36 (1966)
G. Stock, M. Thoss, Phys. Rev. Lett. 78, 578 (1997)
G. Stock, M. Thoss, Phys. Rev. A 59, 64 (1999)
W.H. Miller, C.W. McCurdy, J. Chem. Phys. 69, 5163 (1978)
C.W. McCurdy, H.D. Meyer, W.H. Miller, J. Chem. Phys. 70, 3177 (1979)
R.P. Feynman, Rev. Mod. Phys. 20, 367 (1948)
H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, and Financial Markets (Oxford University Press, Oxford, 2004)
L.S. Schulman, Techniques and Applications of Path Integration (Dover Books on Physics) (Dover Publications, 2005)
Q. Shi, E. Geva, J. Phys. Chem. A 107, 9059 (2003)
J.A. Poulsen, G. Nyman, P.J. Rossky, J. Chem. Phys. 119, 12179 (2003)
J. Liu, W.H. Miller, J. Chem. Phys. 131, 074113 (2009)
J. Beutier, D. Borgis, R. Vuilleumier, S. Bonella, J. Chem. Phys. 141, 084102 (2014)
V.S. Filinov, Nucl. Phys. B 271, 717 (1986)
N. Makri, W.H. Miller, Chem. Phys. Lett. 139, 10 (1987)
M.S. Causo, G. Ciccotti, S. Bonella, R. Vuilleumier, J. Phys. Chem. B 110, 3638 (2006)
M. Monteferrante, S. Bonella, G. Ciccotti, Molec. Phys. 109, 3015 (2011)
D. Mac Kernan, G. Ciccotti, R. Kapral, J. Chem. Phys. 116, 2346 (2002)
C.H. Mak, D. Chandler, Phys. Rev. A 44, 2352 (1991)
M. Topaler, N. Makri, J. Chem. Phys. 101, 7500 (1994)
R. Egger, C.H. Mak, Phys. Rev. B 50, 15210 (1994)
K. Thompson, N. Makri, Chem. Phys. Lett. 291, 101 (1998)
X. Sun, H.B. Wang, W.H. Miller, J. Chem. Phys. 109, 7064 (1998)
K. Thompson, N. Makri, J. Chem. Phys. 110, 1343 (1999)
A. Golosov, D.R. Reichman, J. Chem. Phys. 114, 1065 (2001)
E.R. Dunkel, S. Bonella, D.F. Coker, J. Chem. Phys. 129, 114106 (2008)
P. Huo, D.F. Coker, J. Chem. Phys. 133, 184108 (2010)
P. Huo, T.F. Miller, D.F. Coker, J. Chem. Phys. 139, 151103 (2013)
S. Bonella, M. Monteferrante, C. Pierleoni, G. Ciccotti, J. Chem. Phys. 133, 164105 (2010)
J.J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bonella, S., Ciccotti, G. An introduction to the problem of bridging quantum and classical dynamics. Eur. Phys. J. Spec. Top. 224, 2305–2320 (2015). https://doi.org/10.1140/epjst/e2015-02413-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2015-02413-0