Skip to main content
SpringerLink
Log in

Trajectory Based Simulations of Quantum-Classical Systems

  • Chapter
  • First Online:
Energy Transfer Dynamics in Biomaterial Systems

Part of the book series: Springer Series in Chemical Physics ((CHEMICAL,volume 93))

Abstract

In this Chapter we review the core ingredients of a class of mixed quantum-classical methods that can naturally account for quantum coherence effects. In general, quantum-classical schemes partition degrees of freedom between a quantum subsystem and an environment. The various approaches are based on different approximations to the full quantum dynamics, in particular in the way they treat the environment. Here we compare and contrast two such methods: the Quantum Classical Liouville (QCL) approach, and the Iterative Linearized Density Matrix (ILDM) propagation scheme. These methods are based on evolving ensembles of surface-hopping trajectories in which the ensemble members carry weights and phases and their contributions to time-dependent quantities must be added coherently to approximate interference effects. The side by side comparison we offer highlights similarities and differences between the two approaches and serves as a starting point to explore more fundamental connections between such methods. The methods are applied to compute the evolution of the density matrix of a challenging condensed phase model system in which coherent dynamics plays a critical role: the asymmetric spin-boson. Various implementation questions are addressed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Tully JC. Mixed quantum-classical dynamics: mean-field and surface-hopping. In Classical and Quantum Dynamics in Condensed Phase Simulations, ed. B.J. Berne, G. Ciccotti, D.F. Coker. Chapter 21. Singapore: World Scientific, 1998.

    Google Scholar 

  2. For a review with references to the literature on this topic, see, R. Kapral. Progress in the theory of mixed quantum-classical dynamics. Annu. Rev. Phys. Chem., 57:129, 2006.

    Article  CAS  Google Scholar 

  3. J. C. Tully. Molecular dynamics with electronic transitions. J. Chem. Phys., 93(2):1061, 1990.

    Article  CAS  Google Scholar 

  4. Q. Shi and E. Geva. A derivation of the mixed quantum-classical Liouville equation from the influence functional formalism. J. Chem. Phys., 121(8):3393, 2004.

    Article  CAS  Google Scholar 

  5. S. Nielsen, R. Kapral, and G. Ciccotti. Statistical mechanics of quantum-classical systems. J. Chem. Phys., 115(13):5805, 2001.

    Article  CAS  Google Scholar 

  6. H. Kim and R. Kapral. Transport properties of quantum-classical systems. J. Chem. Phys., 122:214105, 2005.

    Article  Google Scholar 

  7. G. Ciccotti, D.F. Coker, and R.Kapral, Quantum statistical dynamics with trajectories, in Quantum dynamics of complex molecular systems, David Micha and Irene Burghardt, Chemical Physics series vol. 83, (Springer, Berlin), p. 275, 2006.

    Google Scholar 

  8. H. Kim and R. Kapral. Nonadiabatic quantum-classical reaction rates with quantum equilibrium structure. J. Chem. Phys., 123:194108, 2005.

    Article  Google Scholar 

  9. R. Grunwald, H. Kim and R. Kapral. Surface hopping and decoherence with quantum equilibrium structure. J. Chem. Phys., 128:164110, 2008.

    Article  Google Scholar 

  10. G. Hanna and R. Kapral. Quantum-classical Liouville dynamics of nonadiabatic proton transfer. J. Chem. Phys., 122(24):244505, 2005.

    Article  Google Scholar 

  11. G. Hanna and R. Kapral. Quantum-classical Liouville dynamics of proton and deutron transfer rates in a hydrogen bonded complex. J. Chem. Phys., 128:164520, 2008.

    Article  Google Scholar 

  12. R. Kapral and G. Ciccotti. Mixed quantum-classical dynamics. J. Chem. Phys., 110:8919, 1999.

    Article  CAS  Google Scholar 

  13. I. V. Aleksandrov. The statistical dynamics of a system consisting of a classical and a quantum subsystem. Z. Naturforsch., 36:902, 1981.

    Google Scholar 

  14. V. I. Gerasimenko. Correlation-less equations of motion of quantum-classical systems. Repts. Acad. Sci. Ukr.SSR, (10):64, 1981.

    Google Scholar 

  15. V. I. Gerasimenko. Dynamical equations of quantum-classical systems. Theor. Math. Phys., 50:49, 1982.

    Article  Google Scholar 

  16. W. Boucher and J. Traschen. Semiclassical physics and quantum fluctuations. Phys. Rev. D, 37(12):3522, 1988.

    Article  CAS  Google Scholar 

  17. W. Y. Zhang and R. Balescu. Statistical mechanics of a spin polarized plasma. J. Plasma Physics, 40:199, 1988.

    Article  Google Scholar 

  18. Y. Tanimura and S. Mukamel. Multistate quantum Fokker-Planck approach to nonadiabatic wave packet dynamics in pump-probe spectroscopy. J. Chem. Phys., 101:3049, 1994.

    Article  CAS  Google Scholar 

  19. C. C. Martens and J. Y. Fang. Semiclassical-limit molecular dynamics on multiple electronic surfaces. J. Chem. Phys., 106(12):4918, 1997.

    Article  CAS  Google Scholar 

  20. I. Horenko, C. Salzmann, B. Schmidt, and C. Schutte. Quantum-classical liouville approach to molecular dynamics: Surface hopping gaussian phase-space packets. J. Chem. Phys., 117(24):11075, 2002.

    Article  CAS  Google Scholar 

  21. R. Grunwald, A. Kelly and R. Kapral. Quantum dynamics in almost classical environments. this volume, 2009.

    Google Scholar 

  22. D. Mac Kernan, G. Ciccotti, and R. Kapral. Trotter-based simulation of quantum-classical dynamics. J. Phys. Chem. B, 112:424, 2008.

    Article  CAS  Google Scholar 

  23. R. Kapral and G. Ciccotti, A Statistical Mechanical Theory of Quantum Dynamics in Classical Environments, in Bridging Time Scales: Molecular Simulations for the Next Decade, eds. P. Nielaba, M. Mareschal, G. Ciccotti, (Springer, Berlin), p. 445, 2002.

    Google Scholar 

  24. R. Hernandez and G. Voth. Quantum time correlation functions and classical coherence. Chem. Phys., 223:243, 1998.

    Article  Google Scholar 

  25. J.A. Poulsen and G. Nyman and P.J. Rossky. Practical evaluation of condensed phase quantum correlation functions: A Feynman-Kleinert variational linearized path integral method. J. Chem. Phys., 119:12179, 2003.

    Article  CAS  Google Scholar 

  26. Q. Shi and E. Geva. Vibrational energy relaxation in liquid oxygen from a semiclassical molecular dynamics simulation. J. Phys. Chem. A, 107:9070, 2003.

    Article  CAS  Google Scholar 

  27. Q. Shi and E. Geva. Semiclassical theory of vibrational energy relaxation in the condensed Phase. J. Phys. Chem. A, 107:9059, 2003.

    Article  CAS  Google Scholar 

  28. Q. Shi and E. Geva. Nonradiative electronic relaxation rate constants from approximations based on linearizing the path-integral forward-backward action. J. Phys. Chem. A, 108:6109, 2004.

    Article  CAS  Google Scholar 

  29. X. Sun and W.H. Miller. Semiclassical initial value representation for electronically nonadiabatic molecular dynamics. J. Chem. Phys., 106:6346, 1997.

    Article  CAS  Google Scholar 

  30. G. Stock and M. Thoss. Semiclassical description of nonadiabatic quantum dynamics. Phys. Rev. Lett., 78:578, 1997.

    Article  CAS  Google Scholar 

  31. G. Stock and M. Thoss. Mapping approach to the semiclassical description of nonadiabatic quantum dynamics. Phys. Rev. A, 59:64, 1999.

    Article  Google Scholar 

  32. S. Bonella and D.F. Coker. A semi-classical limit for the mapping Hamiltonian approach to electronically non-adiabatic dynamics. J. Chem. Phys., 114:7778, 2001.

    Article  CAS  Google Scholar 

  33. S. Bonella and D.F. Coker. Semi-classical implementation of the mapping Hamiltonian approach for non-adiabatic dynamics: Focused initial distribution sampling. J. Chem. Phys., 118:4370, 2003.

    Article  CAS  Google Scholar 

  34. S. Bonella and D.F. Coker. LAND-map, a linearized approach to nonadiabatic dynamics using the mapping formalism. J. Chem. Phys., 122:194102, 2005.

    Article  CAS  Google Scholar 

  35. S. Bonella, D. Montemayor, and D.F. Coker. Linearized path integral approach for calculating nonadiabatic time correlation functions. Proc. Natl. Acad. Sci., 102:6715, 2005.

    Article  CAS  Google Scholar 

  36. D.F. Coker and S. Bonella, Linearized path integral methods for quantum time correlation functions, in Computer simulations in condensed matter: From materials to chemical biology, eds. M. Ferrario and G. Ciccotti and K. Binder, Lecture Notes in Physics 703, (Springer-Verlag, Berlin), p. 553, 2006.

    Google Scholar 

  37. E. Dunkel, S. Bonella, and D.F. Coker. Iterative linearized approach to non-adiabatic dynamics. J. Chem. Phys., 129:114106, 2008.

    Article  CAS  Google Scholar 

  38. Z. Ma and D.F. Coker. Quantum initial condition sampling for linearized density matrix dynamics: Vibrational pure dephasing of iodine in krypton matrices. J. Chem. Phys., 128:244108, 2008.

    Article  CAS  Google Scholar 

  39. D. Mac Kernan, G. Ciccotti, and R. Kapral. Surface-hopping dynamics of a spin-boson system. J. Chem. Phys., 116(6):2346, 2002.

    Article  CAS  Google Scholar 

  40. H. Kim, A. Nassimi, and R. Kapral. Quantum-classical Liouville dynamics in the mapping basis. J. Chem. Phys., 129:084102, 2008.

    Article  Google Scholar 

  41. D. E. Makarov and N. Makri, Chem. Phys. Lett., 221:482, 1994.

    Article  CAS  Google Scholar 

  42. K. Thompson and N. Makri. Rigorous forward-backward semiclassical formulation of many-body dynamics. Phys. Rev. E, 59(5):R4729, 1999.

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Fisica, Università “La Sapienza”, Piazzale Aldo Moro, 2, 00185, Roma, Italy

    S. Bonella & G. Ciccotti

  2. Department of Chemistry, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts, 02215, U.S.A.

    D. F. Coker

  3. School of Physics, University College Dublin, Belfield, Dublin, 4, Ireland

    D. Mac Kernan

  4. Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario, M5S3H6, Canada

    R. Kapral

Authors
  1. S. Bonella
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. F. Coker
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. Mac Kernan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. Kapral
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. G. Ciccotti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Bonella .

Editor information

Editors and Affiliations

  1. Department de chimie, UMR 8642 due CNRS, Ecole Normale Superieure, rue Lhomond 24, Paris CX 05, 75231, France

    Irene Burghardt

  2. Institut für Physik, AG Halbleitertheorie, Humboldt-Univ. Berlin, Newtonstr. 15, Berlin, 12489, Germany

    V. May

  3. Physical Chemistry Division, University Florida, New Physics Building 2318, Gainesville, 32611-8435, USA

    David A. Micha

  4. Dept. Chemistry, University of Houston, Houston, 77204-5003, USA

    E. R. Bittner

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bonella, S., Coker, D.F., Kernan, D.M., Kapral, R., Ciccotti, G. (2009). Trajectory Based Simulations of Quantum-Classical Systems. In: Burghardt, I., May, V., Micha, D., Bittner, E. (eds) Energy Transfer Dynamics in Biomaterial Systems. Springer Series in Chemical Physics, vol 93. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02306-4_13

Download citation

Publish with us

Policies and ethics

Search

Navigation