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Quantum Dynamics of Small Systems using Discrete Variable Representations

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Supercomputer Algorithms for Reactivity, Dynamics and Kinetics of Small Molecules

Part of the book series: NATO ASI Series ((ASIC,volume 277))

Abstract

The discrete variable representation (DVR) on Gaussian quadrature points for orthogonal polynomials is defined. It is shown that the Hamiltonian for multidimensional systems is easy to evaluate and sparse in the DVR. Methods of solution of the time dependent Schrödinger equation in the DVR are presented. A highly efficient method of solution of the time independent Schrödinger equation for many eigenvalues and eigenvectors is presented which is based (in the DVR) on sequential adiabatic-sudden partitioning diagonalization, truncation, and recoupling of lower dimensional solutions. Brief summaries are presented of the application of these techniques to predissociation of NaI, vibrational states of H3+, and the thermal rate constant of the H+H2 reaction.

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References

  1. R. D. Levine and R. B. Bernstein, Molecular Reaction Dynamics and Chemical Reactivity, 2nd ed. (Clarendon, Oxford, 1987).

    Google Scholar 

  2. R. B. Bernstein, Ed., Atom-Molecule Collision Theory (Plenum, New York, 1979).

    Google Scholar 

  3. W. H. Miller, Ed., Dynamics of Molecular Collisions (Plenum, New York, 1976).

    Google Scholar 

  4. D. C. Clary, Ed., The Theory of Chemical Reaction Dynamics (Reidel, New York, 1986).

    Google Scholar 

  5. D. H. Levy, Ann. Rev. Phys. Chem. 31, 197 (1980).

    Article  CAS  Google Scholar 

  6. J. A. Beswick and J. Jortner, Adv. Chem. Phys. 47, 363 (1981).

    Article  CAS  Google Scholar 

  7. G. Scoles, Ed., Atomic and Molecular Beam Methods (Oxford University Press, London, 1987)

    Google Scholar 

  8. M. Dantus, M. J. Rosker, and A. H. Zewail, J. Chem. Phys. 87, 2395 (1987).

    Article  CAS  Google Scholar 

  9. M. Karplus, R. Porter, and R. D. Sharma, J. Chem. Phys. 49, 3259 (1965).

    Article  Google Scholar 

  10. R. N. Porter and L. M. Raff, in Dynamics of Molecular Collisions, Part A, W. H. Miller, Ed. (Plenum, New York, 1976).

    Google Scholar 

  11. J. C. Tully, Acc. Chem. Res. 14, 188 (1981).

    Article  CAS  Google Scholar 

  12. C. H. Mak, H. C. Andersen, and S. M. George, J. Chem. Phys. 88, 4052 (1988).

    Article  CAS  Google Scholar 

  13. R. E. Cline, Jr., and P. G. Wolynes, J. Chem. Phys. 88, 4334 (1988).

    Article  CAS  Google Scholar 

  14. R. D. Coalson, D. L. Freeman, and J. D. Doll, J. Chem. Phys. 85, 4567 (1986).

    Article  CAS  Google Scholar 

  15. E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular Vibrations (McGraw-Hill, New York, 1955).

    Google Scholar 

  16. I. P. Hamilton and J. C. Light, J. Chem. Phys. 84, 306 (1986).

    Article  CAS  Google Scholar 

  17. I. P. Hamilton, J. Chem. Phys. 87, 774 (1987).

    Article  CAS  Google Scholar 

  18. Z. Bacic and J. C. Light, J. Chem. Phys. 85, 4594 (1986); ibid. 86, 3065 (1987).

    Article  CAS  Google Scholar 

  19. Z. Bacic, D. Watt, and J. C. Light, J. Chem. Phys. 89, 947 (1988).

    Article  CAS  Google Scholar 

  20. J. C. Light and Z. Bacic, J. Chem. Phys. 87, 4008 (1987).

    Article  CAS  Google Scholar 

  21. J. V. Lill, G. A. Parker, and J. C. Light, Chem. Phys. Lett. 89, 483 (1982).

    Article  CAS  Google Scholar 

  22. J. C. Light, I. P. Hamilton, and J. V. Lill, J. Chem. Phys. 82, 1400 (1985); Z. Bacic, R. M. Whitnell, D. Brown, and J. C. Light, Comp. Phys. Commun. (in press).

    Article  CAS  Google Scholar 

  23. A. Askar and A. Cakmak, J. Chem. Phys. 68, 2794 (1978).

    Article  CAS  Google Scholar 

  24. T. J. Park and J. C. Light, J. Chem. Phys. 85, 5870 (1986).

    Article  CAS  Google Scholar 

  25. S. E. Choi and J. C. Light, private communication.

    Google Scholar 

  26. R. M. Whitnell, Ph.D. Thesis, Univ. of Chicago, 1988 (submitted); R. M. Whitnell and J. C. Light, J. Chem. Phys. (in press).

    Google Scholar 

  27. T. J. Park, Ph.D. Thesis, Univ. of Chicago, 1988 (submitted); T. J. Park and J. C. Light, private communication.

    Google Scholar 

  28. D. Gottlieb and S. Orzag, Numerical Analysis of Spectral Methods, Theory and Applications (SIAM, Philadelphia PA, 1977).

    Google Scholar 

  29. L. Fox and I. B. Parker, Chebyshev Polynomials in Numerical Analysis (Oxford University Press, London, 1968); for recent applications in chemical physics, see R. Friesner, J. Chem. Phys. 85, 1462 (1986); ibid. 86, 3522 (1987).

    Google Scholar 

  30. L. Fox, in Methods of Numerical Approximation, D. C. Hanscomb, Ed. (Pergamon, New York, 1966).

    Google Scholar 

  31. R. A. Friesner, J. Phys. Chem. 92, 3091 (1988).

    Article  CAS  Google Scholar 

  32. T. S. Rose, M. J. Rosker, and A. H. Zewail, J. Chem. Phys. 88, 6672 (1988)

    Article  CAS  Google Scholar 

  33. M. J. Rosker, T. S. Rose, and A. H. Zewail, Chem. Phys. Lett 146, 175 (1988).

    Article  CAS  Google Scholar 

  34. W. H. Miller, J. Chem. Phys. 61, 1823 (1974).

    Article  CAS  Google Scholar 

  35. W. H. Miller, S. C. Schwartz, and J. W. Tromp, J. Chem. Phys. 79, 4889 (1983).

    Article  CAS  Google Scholar 

  36. K. Yamashita and W. H. Miller, J. Chem. Phys. 82, 5475 (1985).

    Article  CAS  Google Scholar 

  37. See, for example, C. Lanczos, Applied Analysis (Prentice-Hall, Englewood Cliffs, New Jersey, 1956).

    Google Scholar 

  38. D. O. Harris, G. G. Engerholm, and W. D. Gwinn, J. Chem. Phys. 43, 1515 (1965)

    Article  Google Scholar 

  39. P. F. Endres, J. Chem. Phys. 47, 798 (1967).

    CAS  Google Scholar 

  40. A. S. Dickenson and P. R. Certain, J. Chem. Phys. 49, 4209 (1968) (see, also, ref. 18).

    Article  Google Scholar 

  41. R. W. Heather and J. C. Light, J. Chem. Phys. 79, 147 (1983)

    Article  CAS  Google Scholar 

  42. R. M. Whitnell and J. C. Light, J. Chem. Phys. 86, 2007 (1987)

    Article  CAS  Google Scholar 

  43. J. V. Lill, G. A. Parker, and J. C. Light, J. Chem. Phys. 85, 900 (1986).

    Article  CAS  Google Scholar 

  44. B. Shizgal and R. Blackmore, J. Comput. Phys. 55, 313 (1984);

    Article  Google Scholar 

  45. R. Blackmore and B. Shizgal, Phys. Rev. A31, 1855 (1985).

    Google Scholar 

  46. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Nat. Bur. Std. Applied Mathematics Series 55 (U. S. Govt Printing Office, Washington, D.C., 1964)

    Google Scholar 

  47. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Nat. Bur. Std. Applied Mathematics Series 55 (U. S. Govt Printing Office, Washington, D.C., 1964); P. Dennery and A. Krzywicki, Mathematics for Physicists (Harper & Row, New York, 1967).

    Google Scholar 

  48. M. D. Feit and J. A. Fleck, J. Chem. Phys. 78, 301 (1982); 80, 2578 (1984).

    Article  Google Scholar 

  49. R. Kosloff and R. Kosloff, J. Chem. Phys. 79, 1823 (1983).

    Article  CAS  Google Scholar 

  50. D. Kosloff and R. Kosloff, J. Comput Phys. 52, 35 (1983)

    Article  CAS  Google Scholar 

  51. R. Kosloff and C. Cerjan, J. Chem. Phys. 81, 3722 (1984).

    Article  CAS  Google Scholar 

  52. D. Kouri and R. C. Mowrey, J. Chem. Phys. 86, 2087 (1987); 84, 6466 (1986).

    Article  CAS  Google Scholar 

  53. G. Wahnstrom and H. Metiu, Chem. Phys. Lett 134, 531 (1987)

    Article  CAS  Google Scholar 

  54. B. Jackson and H. Metiu, J. Chem. Phys. 86, 1026 (1987).

    Article  CAS  Google Scholar 

  55. R. Heather and H. Metiu, J. Chem. Phys. 87, 5497 (1987).

    Article  Google Scholar 

  56. R. Heather and H. Metiu, J. Chem. Phys. 86, 5009 (1987)

    Article  CAS  Google Scholar 

  57. G. Jolicard, C. Le Forestier, and E. J. Austin, J. Chem. Phys. 88, 1026 (1988).

    Article  CAS  Google Scholar 

  58. P. Pechukas and J. C. Light, J. Chem. Phys. 44, 3897 (1966).

    Article  Google Scholar 

  59. H. Tal-Ezar and R. Kosloff, J. Chem. Phys. 81, 3967 (1984).

    Article  Google Scholar 

  60. T. J. Park and J. C. Light, J. Chem. Phys. 85, 5870 (1986).

    Article  CAS  Google Scholar 

  61. T. Oka, Phys. Rev. Lett 45, 531 (1980).

    Article  CAS  Google Scholar 

  62. J. K. G. Watson, S. C. Foster, A. R. W. McKellar, P. Bernath, T. Amano, F. S. Pan, M. W. Crofton, R. S. Altman, and T. Oka, Can. J. Phys. 62, 1875 (1984)

    Article  CAS  Google Scholar 

  63. W. A. Majewski, M. D. Marshall, A. R. W. McKellar, J. C. W. Johns, and J. K. G. Watson, J. Mol. Spec. 122, 341 (1987).

    Article  CAS  Google Scholar 

  64. B. Rehfuss, M. Bawendi, and T. Oka, private communication.

    Google Scholar 

  65. C. E. Dykstra and W. C. Swope, J. Chem. Phys. 70, 1 (1979).

    Article  CAS  Google Scholar 

  66. W. Meyer, P. Botschwina, and P. G. Burton, J. Chem. Phys. 84, 891 (1986).

    Article  CAS  Google Scholar 

  67. S. Miller and J. Tennyson, J. Mol. Spec. 128, 132, 530 (1988).

    Article  CAS  Google Scholar 

  68. G. D. Carney and R. N. Porter, J. Chem. Phys. 65, 3547 (1976).

    Article  CAS  Google Scholar 

  69. R. T. Pack, Chem. Phys. Lett. 108, 333 (1984)

    Article  CAS  Google Scholar 

  70. R. T. Pack and G. A. Parker, J. Chem. Phys. 87, 3888 (1987).

    Article  CAS  Google Scholar 

  71. F. Webster and J. C. Light, J. Chem. Phys. 85, 4744 (1986); F. Webster, Ph.D. Thesis, Univ. of Chicago, 1988.

    Article  CAS  Google Scholar 

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Authors and Affiliations

  1. James Franck Institute, Department of Chemistry, University of Chicago, Chicago, Illinois, 60637, USA

    J. C. Light, R. M. Whitnell, T. J. Park & S. E. Choi

Authors
  1. J. C. Light
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  2. R. M. Whitnell
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  3. T. J. Park
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  4. S. E. Choi
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Editor information

Editors and Affiliations

  1. Department of Chemistry, University of Perugia, Perugia, Italy

    Antonio Laganà

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© 1989 Kluwer Academic Publishers

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Light, J.C., Whitnell, R.M., Park, T.J., Choi, S.E. (1989). Quantum Dynamics of Small Systems using Discrete Variable Representations. In: Laganà, A. (eds) Supercomputer Algorithms for Reactivity, Dynamics and Kinetics of Small Molecules. NATO ASI Series, vol 277. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0945-8_11

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  • DOI: https://doi.org/10.1007/978-94-009-0945-8_11

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6915-1

  • Online ISBN: 978-94-009-0945-8

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