Reactive Scattering Resonances and their Physical Interpretation: The Vibrational Structure of the Transition State

  • Aron Kuppermann

Abstract

Quantum mechanical structure in reaction-probability-versus-energy curves for a realistic potential energy surface was first observed about a decade ago,1,2 for the collinear H + H2 system. Such structure had been previously found for a potential energy surface having sharp edges,3 made of piecewise-constant potentials, but in one-mathematical-dimensional (1MD) barrier problems, structure in transmission-probability-versus-energy curves4 is known to disappear when a rectangular barrier is replaced by one which is sufficiently “rounded”, such as parabolic5,6 or Eckart6,7 barriers, and is attributed to edge diffraction effects. The structure in the collinear H + H2 results on a smoothly varying surface, at energies above the vibrational excitation threshold of reaction products, was guessed1b as being due to interference effects between different reaction paths, a guess subsequently confirmed by a quantum mechanical lifetime analysis8 and a semiclassical calculation.9 The former indicated the concomitant presence of and interference between direct and dynamic resonance (Feshbach10) processes. The mechanism of such resonances was attributed to the existence of wells in the vibrationally adiabatic potentials along the minimum energy path.2b

Keywords

Potential Energy Surface Reaction Probability Translational Energy Exact Quantum Minimum Energy Path 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.(a)
    D. G. Truhlar and A. Kuppermann, Quantum mechanics of the H + H2 reaction: Exact scattering probabilities for collinear collisions, J. Chem. Phys. 52: 3841 (1970).CrossRefGoogle Scholar
  2. (b).
    D. G. Truhlar and A. Kuppermann, Exact and approximate quantum mechanical reaction probabilities and rate constants for the collinear H + H2 reaction, J. Chem. Phys. 56: 2232 (1972).CrossRefGoogle Scholar
  3. 2.(a)
    S.-F. Wu and R. D. Levine, Quantum mechanical computational studies of chemical reactions: I. Close-coupling method for the collinear H + H2 reaction, Mol. Phys. 22: 881 (1971).CrossRefGoogle Scholar
  4. (b).
    R. D. Levine and S.-F. Wu, Resonances in reactive collisions: Computational study of the H + H2 collision, Chem. Phys. Lett, 11: 557 (1971).CrossRefGoogle Scholar
  5. 3.
    K. T. Tang, B. Kleinman, and M. Karplus, Solvable quantum-mechanical model of three-body rearrangement scattering, J. Chem. Phys. 50: 1119 (1969).CrossRefGoogle Scholar
  6. 4.
    L. I. Schiff, “Quantum Mechanics”, 3rd ed., McGraw-Hill, New York (1968), pp. 102–104.Google Scholar
  7. 5.(a)
    R. P. Bell, The tunnel effect correction for parabolic potential barriers, Trans. Faraday Soc 55: 1 (1959).CrossRefGoogle Scholar
  8. (b).
    H. S. Johnston, “Gas Phase Reaction Rate Theory”, The Ronald Press, New York (1966), pp. 40–45.Google Scholar
  9. 6.
    H. Jeffreys and B. Jeffreys, “Methods of Mathematical Physics”, 3rd ed., Cambridge University Press, New York (1962), p. 703.Google Scholar
  10. 7.
    C. Eckart, The penetration of a potential barrier by electrons, Phys. Rev. 35: 1303 (1930).CrossRefGoogle Scholar
  11. 8.
    G. C. Schatz and A. Kuppermann, Role of direct and resonant (compound state) processes and of their interferences in the quantum dynamics of the collinear H + H2 exchange reaction, J. Chem. Phys. 59: 964 (1973).CrossRefGoogle Scholar
  12. 9.
    J. R. Stine and R. A. Marcus, Semiclassical S matrix theory for a compound state resonance in the reactive collinear H + H2 collision, Chera. Phys. Lett. 29: 575 (1974).CrossRefGoogle Scholar
  13. 10.
    H. Feshbach, Unified theory of nuclear reactions, Ann. Phys. NY 5: 357 (1958).CrossRefGoogle Scholar
  14. H. Feshbach, A unified theory of nuclear reactions. II, Ann. Phys. NY 19: 287 (1962).CrossRefGoogle Scholar
  15. 11.(a)
    G. C. Schatz, J. M. Bowman, and A. Kuppermann, Large quantum effects in the collinear F + H2 → FH + H reaction, J. Chem. Phys. 58: 4023 (1973).CrossRefGoogle Scholar
  16. (b).
    G. C. Schatz, J. M. Bowman, and A. Kuppermann, Exact quantum, quasiclassical, and semiclassical reaction probabilities for the collinear F + H2 → FH + H reaction, J. Chem. Phys. 63: 674 (1975).CrossRefGoogle Scholar
  17. (c).
    G. C. Schatz, J. M. Bowman, and A. Kuppermann, Exact quantum, quasi-classical, and semiclassical reaction probabilities for the collinear F + D2 → FD + D reaction, J. Chem. Phys. 63: 685 (1975).CrossRefGoogle Scholar
  18. 12.
    S. F. Wu, B. R. Johnson, and R. D. Levine, Quantum mechanical computational studies of chemical reactions: III. Collinear A + BC reaction with some model potential energy surfaces, Mol. Phys. 25: 839 (1973).CrossRefGoogle Scholar
  19. 13.
    J. N. L. Connor, W. Jakubetz, and J. Manz, Exact quantum mechanical probabilities by the state path sum method: Collinear F + H2 reaction, Mol. Phys. 29: 347 (1975).CrossRefGoogle Scholar
  20. J. N. L. Connor, W. Jakubetz, and J. Manz, The F + H2(v=0) → FH(v′≤3) + H reaction: Quantum collinear reaction probabilities on three different potential energy surfaces, Mol. Phys. 35: 1301 (1978).CrossRefGoogle Scholar
  21. J. N. L. Connor, W. Jakubetz, and J. Manz, Quantum collinear reaction probabilities for vibrationally excited reactants: F + H2(v≤2) → FH(v′≤5) + H, Mol. Phys. 39: 799 (1980).CrossRefGoogle Scholar
  22. B. C. Garrett, D. G. Truhlar, R. S. Grev, A. W. Magnuson, and J. N. L. Connor, Variational transition state theory, vibrationally adiabatic transmission coefficients, and the unified statistical model tested against accurate quantal rate constants for collinear F + H2, H + F2, and isotopic analogs, J. Chem. Phys. 73: 1721 (1980).CrossRefGoogle Scholar
  23. 14.
    R. E. Wyatt, J. A. McNutt, S. L. Latham, and M. J. Redmon, general discussion, Faraday Disc. Chem. Soc. 62: 322 (1977).Google Scholar
  24. R. E. Wyatt, Quantum mechanics of neutral atom-diatomic molecule reactions, in: “State-to-State Chemistry”, P. R. Brooks and E. F. Hayes, eds., American Chemical Society, Washington (1977), p. 185.CrossRefGoogle Scholar
  25. S. L. Latham, J. F. McNutt, R. E. Wyatt, and M. J. Redmon, Quantum dynamics of the F + H2 reaction: Resonance models and energy and flux distributions in the transition state, J. Chem. Phys. 69: 3746 (1978).CrossRefGoogle Scholar
  26. 15.
    M. Baer, An exact quantum mechanical study of the isotopic collinear reactive systems H2 + Cl and D2 + Cl, Mol. Phys. 27: 1429 (1974).CrossRefGoogle Scholar
  27. M. Baer, U. Halavee, and A. Persky, The collinear Cl + XY(X,Y= H,D,T). A comparison between quantum mechanical, classical, and transition state theory results, J. Chem. Phys. 61: 5122 (1974).CrossRefGoogle Scholar
  28. 16.(a)
    F. M. Chapman, Jr. and E. F. Hayes, Resonances in the collinear inelastic scattering of He by \(H_{2}^{+}\) below the reaction threshold, J. Chem. Phys. 62: 4400 (1975).CrossRefGoogle Scholar
  29. (b).
    F. M. Chapman, Jr. and E. F. Hayes, Open and closed channel resonances in the collinear inelastic scattering of He by \(H_{2}^{+}\), J. Chem. Phys. 65: 1032 (1976).CrossRefGoogle Scholar
  30. 17.
    D. J. Kouri and M. Baer, Collinear quantum mechanical calculations of the \(H_{2}^{+}\) proton transfer reaction, Chem. Phys. Lett. 24: 37 (1974).CrossRefGoogle Scholar
  31. 18.
    J. T. Adams, Collinear quantum mechanical calculations for the reaction \(He+H_{2}^{+}\rightarrow HeH^{+}+H\), Chem. Phys. Lett. 33: 275 (1975).CrossRefGoogle Scholar
  32. 19.(a)
    F. M. Chapman, Jr. and E. F. Hayes, Quantum dynamical study of resonance effects in the collinear reaction H2 + I → HI + I, J. Chem.Phys. 66: 2554 (1977).CrossRefGoogle Scholar
  33. (b).
    J. C. Gray, D. G. Truhlar, L. Clemens, J. W. Duff, F. M. Chapman, Jr., G. O. Morrell, and E. F. Hayes, Quasiclassical trajectory calculations compared to quantum mechanical reaction probabilities, rate constants, and activation energies for two different potential energy surfaces for the collinear reaction H2 + I → H + HI, including dependence on initial vibrational state, J. Chem. Phys. 69: 240 (1978).CrossRefGoogle Scholar
  34. 20.
    J. M. Bowman, S. C. Leasure, and A. Kuppermann, Large quantum effects in a model electronically non-adiabatic reaction: Ba + N2O → BaO* + N2, Chem. Phys. Lett. 43: 374 (1976).CrossRefGoogle Scholar
  35. 21.
    J. A. Kaye and A. Kuppermann, Collinear quantum mechanical probabilities for the I + HI → IH + I reaction using hyperspherical coordinates, Chem. Phys. Lett. 77: 573 (1981).CrossRefGoogle Scholar
  36. 22.
    G. C. Schatz and A. Kuppermann, Dynamical resonances in collinear, coplanar, and three-dimensional quantum mechanical reactive scattering, Phys. Rev. Lett. 35: 1266 (1975).CrossRefGoogle Scholar
  37. 23.
    M. J. Redmon and R. E. Wyatt, Quantum resonance structure in the three-dimensional F + H2 reaction, Chem. Phys. Lett. 63: 209 (1979).CrossRefGoogle Scholar
  38. 24.
    R. K. Sparks, C. C. Hayden, K. Shobatake, D. M. Neumark, and Y. T. Lee, Molecular beam studies of reaction dynamics of F + H2, D2, in: “Horizons of Quantum Chemistry”, K. Fukui and B. Pullman, eds., D. Reidel Publishing Co., Boston (1980), p. 91.CrossRefGoogle Scholar
  39. 25.
    R. N. Porter and M. Karplus, Potential energy surface for H3, J. Chem. Phys. 40: 1105 (1964).CrossRefGoogle Scholar
  40. 26.
    J. T. Muckerman, unpublished. See reference 11b.Google Scholar
  41. 27.
    G. C. Schatz and A. Kuppermann, Vibrational deactivation on chemically reactive potential energy surfaces: An exact quantum study of a low barrier collinear model of H + FH, D + FD, H + FD, and D + FH, J. Chem. Phys. 72: 2737 (1980).CrossRefGoogle Scholar
  42. 28.
    G. C. Schatz and A. Kuppermann, An analysis of resonant and direct processes in collinear atom-diatom reactions, manuscript in preparation.Google Scholar
  43. 29.
    A. Kuppermann, J. A. Kaye, and J. P. Dwyer, Hyperspherical coordinates in quantum mechanical collinear reactive scattering, Chem. Phys. Lett. 74: 257 (1980).CrossRefGoogle Scholar
  44. 30.
    E. Merzbacher, “Quantum Mechanics”, 2nd ed., John Wiley & Sons, New York (1970), pp. 108–113.Google Scholar
  45. 31.
    G. Breit and E. Wigner, Capture of slow neutrons, Phys. Rev. 49: 519 (1936).CrossRefGoogle Scholar
  46. 32.
    A. M. Lane and R. G. Thomas, R-matrix theory of nuclear reactions, Rev. Mod. Phys. 30: 257 (1958).CrossRefGoogle Scholar
  47. 33.
    G. C. Schatz, the quantum dynamics of atom plus diatom chemical reactions, Ph.D. thesis, California Institute of Technology, Pasadena, CA, 1976, p. 619.Google Scholar
  48. 34.
    A. Kuppermann, Accurate quantum calculations of reactive systems, Theor. Chem.: Advan. Perspectives 6A: 79 (1981).Google Scholar
  49. 35.
    E. P. Wigner, Lower limit of the energy derivative of the scattering phase shift, Phys. Rev. 98: 145 (1952); L. Eisenbud, Ph.D. thesis, Princeton University, Princeton, NJ, 1948.CrossRefGoogle Scholar
  50. 36.
    F. T. Smith, Lifetime matrix in collision theory, Phys. Rev. 118: 349 (1960).CrossRefGoogle Scholar
  51. 37.
    R. K. Adair, High-energy maxima in the Tr-p cross sections, Phys. Rev. 113: 338 (1959).CrossRefGoogle Scholar
  52. 38.
    R. D. Levine, M. Shapiro, and B. R. Johnson, Transition probabilities in molecular collisions: Computational studies of rotational excitation, J. Chem. Phys. 52: 1755 (1970).CrossRefGoogle Scholar
  53. 39.
    J. A. Kaye and A. Kuppermann, unpublished results.Google Scholar
  54. 40.
    A. Kuppermann and J. A. Kaye, Collision lifetime matrix analysis of the first resonance in the collinear F + H2 reaction and its isotopically substituted analogs, Chem. Phys. Lett., submitted for publication.Google Scholar
  55. 41.
    V. K. Babamov and A. Kuppermann, A physical interpretation of the collinear reactive scattering resonances in the F + H2, HD, DH, and D2 systems, manuscript in preparation.Google Scholar
  56. 42.
    L. M. Delves, Tertiary and general-order collisions, Nucl. Phys. 9: 391 (1959).Google Scholar
  57. L. M. Delves, Tertiary and general-order collisions (II), Nucl. Phys. 20: 275 (1960).CrossRefGoogle Scholar
  58. 43.
    A. Kuppermann and J. P. Dwyer, A simple model of dynamic resonances in collinear reactive scattering, in: “Electronic and Atomic Collisions, Abstracts of Contributed Papers, XIth International Conference on the Physics of Electronic and Atomic Collisions, Kyoto”, K. Takayanagi and N. Oda, eds., The Society for Atomic Collision Research, Japan (1979), pp. 888-889; J. P. Dwyer and A. Kuppermann, Resonances in collinear reactive scattering: A simple hyperspherical coordinate model, manuscript in preparation.Google Scholar
  59. 44.
    J. M. Bowman, G. C. Schatz, and A. Kuppermann, unpublished calculations. See also.Google Scholar
  60. B. C. Garrett and D. G. Truhlar, Generalized transition state theory. Quantum effects for collinear reactions of hydrogen molecules and isotopically substituted hydrogen molecules, J. Phys. Chem. 83: 1079 (1979).CrossRefGoogle Scholar
  61. B. C. Garrett and D. G. Truhlar, Generalized transition state theory. Quantum effects for collinear reactions of hydrogen molecules and isotopically substituted hydrogen molecules, J. Phys. Chem. 84: 682(E) (1980).Google Scholar
  62. 45.(a)
    A. Kuppermann and G. C. Schatz, Quantum mechanical reactive scattering: An accurate three-dimensional calculation, J. Chem. Phys. 62: 2502 (1975).CrossRefGoogle Scholar
  63. (b).
    G. C. Schatz and A. Kuppermann, Quantum mechanical reactive scattering for three-dimensional atom plus diatom systems. I. Theory, J. Chem. Phys. 65: 4642 (1976).CrossRefGoogle Scholar
  64. (c).
    G. C. Schatz and A. Kuppermann, Quantum mechanical reactive scattering for three-dimensional atom plus diatom systems. II. Accurate cross sections for H + H2, J. Chem. Phys. 65: 4668 (1976).CrossRefGoogle Scholar
  65. 46.
    A. B. Elkowitz and R. E. Wyatt, Quantum mechanical reaction cross sections for the three-dimensional hydrogen exchange reaction, J. Chem. Phys. 62: 2504 (1975).CrossRefGoogle Scholar
  66. 47.
    R. B. Walker, E. B. Stechel, and J. C. Light, Accurate H3 dynamics on an accurate H3 potential energy surface, J. Chem. Phys. 69: 2922 (1978).CrossRefGoogle Scholar
  67. 48.
    H. Ehrhardt, Recent experimental progress in e-H, e-He+, and e-He resonance scattering, in: “Physics of the One and Two Electron Atoms”, F. Bopp and H. Kleinpoppen, eds., North-Holland Publishing Co., Amsterdam (1969), p. 598.Google Scholar

Copyright information

© Springer Science+Business Media New York 1981

Authors and Affiliations

  • Aron Kuppermann
    • 1
  1. 1.Arthur Amos Noyes Laboratory of Chemical PhysicsCalifornia Institute of TechnologyPasadenaUSA

Personalised recommendations