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Quantum-dressed Classical Mechanics

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Abstract

Classical mechanics is known often to offer a good description of molecular dynamical processes and hence this approach is widely used to simulate the dynamics of molecular systems. Classical mechanics allows for simulation of large systems. Large means in this connection systems consisting of several thousands of atoms or molecules. The reason for this is that the computational effort in classical dynamics scales about linearly with the size of the system, i.e. with the number of atoms N. Hence, within the classical mechanical description it is therefore possible to simulate phenomena as for instance protein folding or phase transitions. However, for a number of dynamical processes the classical mechanical description appears to be insufficient. In general this is so for what could be called “rare events”, i.e. processes which have probabilities of the order 10−3 or smaller. Such processes could for instance be reactions where a barrier has to be tunneled through or state resolved vibrational or electronic transitions which are classically forbidden. By classically forbidden we understand processes which for dynamical reasons do not occur in classical mechanics. Thus, for a large class of problems involving for instance chemical reactions with activation barrier, vibrational and electronic transitions we will have to use the correct description, namely the quantum description. However, the quantum mechanical approach has the problem that the computationally effort scales exponentially as 103N, i.e. even today problems with N=3~4 (three to four atoms) can only be solved “exactly” if one or more of the atoms are hydrogen atoms.

Keywords

Grid Point Classical Mechanic Coupling Matrix Classical Trajectory Gaussian Wave Packet 
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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References

  1. 1.
    G.D. Billing, The quantum-classical theory, Oxford University Press, New York 2002.Google Scholar
  2. 2.
    Feynmann, R. P., and Hibbs, A. R., 1965, Quantum Mechanics and Path Integrals, McGraw Hill, New York.Google Scholar
  3. 3.
    Pechukas, P., 1969, Phys. Rev. 181; 166-173, 174.ADSCrossRefGoogle Scholar
  4. 4.
    Miller, W.H., 1970, J. Chem. Phys. 53, 1949, 3578.MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Marcus, R. A., 1970, Chem. Phys. Lett. 7, 525.ADSCrossRefGoogle Scholar
  6. 6.
    van Vleck, J. H., 1928, Proc. Nat. Acad. Sci. 14, 178-188.ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Herman, M. F., and Kluk, E., 1984, Chem. Phys. 91, 27-34.CrossRefADSGoogle Scholar
  8. 8.
    Miller, W. H., 1991, J. Chem. Phys. 95, 9428.ADSCrossRefGoogle Scholar
  9. 9.
    Lebedeff, S. A., 1968,Phys. Rev. 165, 1399.ADSCrossRefGoogle Scholar
  10. 10.
    Heller, E. J., 1975, J. Chem. Phys. 62, 1544-1555ADSCrossRefGoogle Scholar
  11. 10.a
    Heller, E. J., 1975, J. Chem. Phys. 1976, 64, 63; 1977, 66, 5777.MathSciNetADSCrossRefGoogle Scholar
  12. 1 l.
    Sawada, S. I., Heather, R., Jackson, B., and Metiu, H.; 1985, J. Chem. Phys. 83, 3009.ADSCrossRefGoogle Scholar
  13. 12.
    Martinez, T. J., Ben-Nun, M.; and Levine, R. D., 1996, J. Phys. Chem. 100, 7884.CrossRefGoogle Scholar
  14. 13.
    Child, M. S., 1974, Molecular Collision Theory”, Academic Press, London; 1991, Semiclassical Mechanics with Molecular Applications, Clarendon Press, Oxford.Google Scholar
  15. 14.
    Manz, J., 1997, in Femtochemistry and Femtobiology: Ultrafast Reaction Dynamics at Atomic- Scale Resolution, Ed. V. Sundstöm, Imperial College Press, London, pp.80-318.Google Scholar
  16. 15.
    Kay, K. G., 1997, J. Chem. Phys., 107, 2313-2328.ADSCrossRefGoogle Scholar
  17. 16.
    Adhikari S., and Billing, G. D., 2000, J. Chem. Phys. 113, 1409-1414ADSCrossRefGoogle Scholar
  18. 16.a
    Billing, G. D., and Adhikari, S., 2000, Chem. Phys. Lett. 321, 197-204.ADSCrossRefGoogle Scholar
  19. 17.
    Billing, G. D., 2001, Chem. Phys. Lett. 343, 130-138.ADSCrossRefGoogle Scholar
  20. 18.
    Billing, G. D., 2001, J. Chem. Phys. 114, 6641-6653; 2001ADSCrossRefGoogle Scholar
  21. 18.a
    Billing, G. D., 2001 Int. J. Quant. Chem. 84, 467-478.CrossRefGoogle Scholar
  22. 19.
    Billing, G. D., 2001, Chem. Phys. 264, 71-80.ADSCrossRefGoogle Scholar
  23. 20.
    Billing, G. D., 2001, J. Chem. Phys. 105, 2340-2347.CrossRefGoogle Scholar
  24. 21.
    Billing, G. D., 1997, J. Chem. Phys. 107, 4286-4294.ADSCrossRefGoogle Scholar
  25. 22.
    Billing, G. D., 1999, J. Chem. Phys. lll, 48-53.Google Scholar
  26. 23.
    See for instance Coalson, R. D., and Karplus, M., 1982, Chem. Phys. Lett. 90, 301-305ADSCrossRefGoogle Scholar
  27. 23.a
    Meyer, H.-D., 1981, Chem. Phys. 61, 335;ADSCrossRefGoogle Scholar
  28. 23.b
    Kay, K. G., 1992, Phys. Rev. A 46, 1213-1232;MathSciNetADSCrossRefGoogle Scholar
  29. 23.c
    Møller, K. B., and Henriksen N. E., 1996, J. Chem. Phys.105 5037-5047.ADSCrossRefGoogle Scholar
  30. 24.
    Bohm, D., 1952,, Phys. Rev. 85, 166-179; ibid. 180-193.MathSciNetADSCrossRefzbMATHGoogle Scholar
  31. 25.
    Billing, G. D., 2000, J. Mol. Structure (Theochem), 501-502, 519-528; 2001CrossRefGoogle Scholar
  32. 25.a
    Billing, G. D., 2000, Phys. Chem. Chem.. Phys. 1, 4687-4694.CrossRefGoogle Scholar
  33. 26.
    Billing, G. D., 2000, Quantum Classical Methods, in Lecture Notes, Eds. Lagana, A.; Riganelli, A. Springer Verlag; Berlin.Google Scholar
  34. 27.
    See e.g. Cullum, J. K., and Willoughby, R. A., 1985, Lanczos Algorithms for large Symmetrix Eigenvalue Computations, Birkhaüser, Boston.Google Scholar
  35. 28.
    Park, T. J., and Light, J. C., 1986, J. Chem. Phys. 85, 5870-5876.ADSCrossRefGoogle Scholar
  36. 29.
    Billing, G. D., and Mikkelsen, K. V., 1997, Advanced Molecular Dynamics and Chemical Kinetics, Wiley; New York.zbMATHGoogle Scholar
  37. Billing, G. D. unpublished results.Google Scholar
  38. Coletti, C., and Billing, G. D. to be published.Google Scholar
  39. 32.
    Billing, G. D., 2001, Chem. Phys. Lett. 339, 237-242.ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  1. 1.Department of Chemistry, H.C. Ørsted InstituteUniversity of CopenhagenDenmark

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