How to Calculate Rotational and Vibrational Cross Sections for Low-Energy Electron Scattering from Diatomic Molecules using Close-Coupling Techniques

  • Michael A. Morrison
  • Weiguo Sun


This chapter is not a review; electron-molecule dynamics is already replete with fine reviews, many of which appear in books devoted entirely to this topic.1–6 These reviews discuss the applied importance of this field,7–10 survey the status of electron-molecule collision data,11–40 and address specialized topics such as resonance scattering,15 vibrational excitation,16 near-threshold scattering,17–18 particular theoretical approaches such as the R-matrix method,19 numerical methods for solving the Schrödinger equation,20–22 and scattering from polar23,24 and polyatomic25,26 targets. Neither is this chapter primarily pedagogical; readers can find elsewhere a wealth of pedagogically useful tutorial introductions and reviews that narrate the major developments in the field’s long rich history and survey recent advances that have made it the focus of intense activity during the last 20 years.27–30 Rather, this chapter is a “ready reference” of the key equations for the application of one very widely used theoretical strategy—the eigenfunction-expansion or “close-coupling“ method— to one very important class of problems: quantum scattering (at incident energies less than about 10 eV) from a closed-shell diatomic molecule accompanied, perhaps, by rotational and/or vibrational (but not electronic) excitation of the target. As exemplary of an extremely powerful method for reducing multi-variable integro-differential equations to more tractable sets of fewer-variable equations, this class of problems illustrates strategies used in many other theoretical contexts, both within and outside of electron-molecule collisions.


Orbital Angular Momentum Schrodinger Equation Partial Cross Section Simple Harmonic Oscillator Internuclear Separation 
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  1. 1.
    Electron Molecule and Photon-Molecule Collisions, edited by T. N. Rescigno, V. McKoy and B. I. Schneider, (New York, Plenum 1979).Google Scholar
  2. 2.
    Electron-Molecule Collisions and Photoionization Processes, edited by V. McKoy, H. Suzuki, K. Takayanagi, and S. Trajmar (Deerfield Beach, Florida: Verlag Chemie International, 1983).Google Scholar
  3. 3.
    Wavefunctions and Mechanisms for Electron Scattering Processes, edited by F. A. Gianturco and G. Stefani (Springer-Verlag, 1984).Google Scholar
  4. 4.
    Electron-Molecule Interactions and their Applications, Volume 1, edited by L. G. Christophorou (Academic Press, New York, 1984).Google Scholar
  5. 5.
    Electron-Molecule Collisions, edited by I. Shimamura and K. Takayanagi (New York, Plenum, 1984).Google Scholar
  6. 6.
    Swarm Studies and Inelastic Electron-Molecule Collisions, edited by L. C. Pitchford, V. McKoy, A. Chutjian, and S. Trajmar (New York: Springer-Verlag, 1986).Google Scholar
  7. 7.
    G. J. Schulz in Principles of Laser Plasmas, edited by G. Bekite (Wiley, New York, 1976), Chap. 2.Google Scholar
  8. 8.
    A. V. Phelps in Electron-Molecule Scattering, edited by S. C. Brown (Wiley-Interscience, New York, 1979), Chap. 2.Google Scholar
  9. 9.
    J. N. Bardsley, in Electron-Molecule Collisions and Photoionization Processes, edited by V. McKoy, H. Suzuki, K. Takayanagi, and S. Trajmar (Deerfield Beach. Fl: Verlag Chemie International, 1983), p.235.Google Scholar
  10. 10.
    G. S. Willet, Introduction to Gas Lasers—Population Inversion Mechanisms (New York: Pergammon, 1984).Google Scholar
  11. 11.
    S. Trajmar, D. F. Register, and A. J. Chutjian, Phys. Rept. 97, 220 (1983).ADSCrossRefGoogle Scholar
  12. 12.
    J. W. McConkey, S. Trajmar, and G. C. M. King, Comments At. Mol. Phys. 22, 17 (1988).Google Scholar
  13. 13.
    M. Kimura and M. Inokuti, Comments At. Mol. Phys. 24, 269 (1990).Google Scholar
  14. 14.
    W. L. Morgan, Plasma Chemistry and Plasma Processing 12, 449 (1992); JILA Data Center Report No. 34 (1991).CrossRefGoogle Scholar
  15. 15.
    G. J. Schulz, Rev. Mod. Phys. 45, 423 (1962).ADSCrossRefGoogle Scholar
  16. 16.
    D. G. Thompson, Adv. At. Mol. Phys. 19, 309 (1984).ADSCrossRefGoogle Scholar
  17. 17.
    E. Enhardt and L. Frost, Comments At. Mol. Phys. 29, 123 (1993).Google Scholar
  18. 18.
    M. A. Morrison, Adv. At. Mol. Phys. 24, 51 (1988).ADSCrossRefGoogle Scholar
  19. 19.
    P. G. Burke, in Quantum Dynamics of Molecules, ed. by R. G. Wooley (New York: Plenum, 1980), pp. 483.CrossRefGoogle Scholar
  20. 20.
    M. A. Morrison, in Electron-and Photon-Molecule Collisions edited by T. N. Rescigno, B. V. McKoy and B. I. Schneider (Plenum Press, New York, 1979).Google Scholar
  21. 21.
    B. D. Buckley, P. G. Burke, and C. J. Noble in Electron-Molecule Collisions edited by I. Shimamura and K. Takayanagi (Plenum, New York, 1984) page 495.CrossRefGoogle Scholar
  22. 22.
    L. A. Collins and B. I. Schneider, in Electron-Molecule Scattering Processes and Photoionization, edited by P. G. Burke and J. B. West (New York: Plenum, 1988).Google Scholar
  23. 23.
    Y. Itikawa, Phys. Rept. 46, 117 (1978).ADSCrossRefGoogle Scholar
  24. 24.
    L. A. Collins and D. W. Norcross, Adv. At. Mol. Phys. 18, 341 (1983).Google Scholar
  25. 25.
    D. G. Thompson and F. A. Gianturco, Comments At. Mol. Phys. 16, 307 (1985).Google Scholar
  26. 26.
    F. A. Gianturco and A. Jain, Phys. Rept. 143, 347 (1986).ADSCrossRefGoogle Scholar
  27. 27.
    D. E. Golden, N. F. Lane, A. Temkin, and E. Gerjuoy, Rev. Mod. Phys. 43, 642 (1971).ADSCrossRefGoogle Scholar
  28. 28.
    P. G. Burke, Adv. At. Mol. Phys. 15, 471 (1979).ADSCrossRefGoogle Scholar
  29. 29.
    N. F. Lane, Rev. Mod. Phys. 52, 29 (1980).ADSCrossRefGoogle Scholar
  30. 30.
    M. A. Morrison, Aust. J. Phys. 36, 239 (1983).ADSCrossRefGoogle Scholar
  31. 31.
    M. Born and J. R. Oppenheimer, Ann. Phys. (Leipzig) 84, 457 (1927).ADSzbMATHGoogle Scholar
  32. 32.
    M. A. Morrison, T. L. Estle, and N. F. Lane, Quantum States of Atoms, Molecules, and Solids (Prentice-Hall, Englewood Cliffs, New Jersey, 1977).Google Scholar
  33. 33.
    A primary resource for electron-molecule codes is the program library of Computer Physics Communcations. Information is available from Department of Applied Mathematics and Theoretical Physics, The Queen’s University of Belfast, Belfast BT7 INN, Northern Ireland.Google Scholar
  34. 34.
    M. A. Morrison and G. A. Parker, Aust. J. Phys. 40, 465 (1987).ADSCrossRefGoogle Scholar
  35. 35.
    M. E. Rose, Elementary Theory of Angular Momentum (New York, Wiley, 1957).zbMATHGoogle Scholar
  36. 36.
    D. A. Varshalovich, A. N. Moskalev and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).Google Scholar
  37. 37.
    R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics (Wiley, New York, 1988).Google Scholar
  38. 38.
    D. M. Brink and G. R. Satchler, Angular Momentum (Third Edition) (New York: Oxford, 1993).Google Scholar
  39. 39.
    B. I. Schneider and L. A. Collins, J. Phys. B: At. Mol. Phys. 15, L335 (1982); Phys. Rev. A 27, 2847 (1983).ADSCrossRefGoogle Scholar
  40. 40.
    H. D. Meyer, J. Phys. B: At. Mol. Phy. 25, 2657 (1992).ADSCrossRefGoogle Scholar
  41. 41.
    T. L. Gibson and M. A. Morrison, Phys. Rev. A 29, 2497 (1984).ADSCrossRefGoogle Scholar
  42. 42.
    J. K. O’Connell and N. F. Lane, Phys. Rev. A 27, 1893 (1983).ADSCrossRefGoogle Scholar
  43. 43.
    A. Jain and D. W. Norcross, Phys. Rev. A 34, 739 (1986).Google Scholar
  44. 44.
    M. A. Morrison and W. K. Trail, Phys. Rev. A 48, 2874 (1993).ADSCrossRefGoogle Scholar
  45. 45.
    M. J. Seaton, Comments At. Mol. Phys. 1, 184 (1970).ADSGoogle Scholar
  46. 46.
    M. A. Morrison, and L. A. Collins, Phys. Rev. A 23, 127 (1981).ADSCrossRefGoogle Scholar
  47. 47.
    L. A. Collins, W. D. Robb, and M. A. Morrison, Phys. Rev. A 21, 488 (1980).ADSCrossRefGoogle Scholar
  48. 48.
    B. I. Schneider and L. A. Collins, Comput. Phys. Rpt. 10, 51 (1989).ADSGoogle Scholar
  49. 49.
    E. R. Cohen and B. N. Taylor, Rev. Mod. Phys. 59, 1121 (1987).ADSCrossRefGoogle Scholar
  50. 50.
    N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions (Third Edition) (Oxford: Clarendon Press, 1965).Google Scholar
  51. 51.
    J. R. Taylor, Scattering Theory, (New York: Wiley, 1972).Google Scholar
  52. 52.
    S. Hara, J. Phys. Soc. Jpn. 27, 1592 (1969).ADSCrossRefGoogle Scholar
  53. 53.
    A. Temkin and K. V. Vasavada, Phys. Rev. A 160, 190 (1967).Google Scholar
  54. 54.
    N. Chandra and A. Temkin, Phys. Rev. A 13, 188 (1976).ADSCrossRefGoogle Scholar
  55. 55.
    M. A. Morrison, A. N. Feldt, and D. A. Austin, Phys. Rev. A 29, 2518 (1984).ADSCrossRefGoogle Scholar
  56. 56.
    A. N. Feldt and M. A. Morrison, Phys. Rev. A 29, 401 (1984).ADSCrossRefGoogle Scholar
  57. 57.
    M. A. Morrison, Understanding Quantum Physics: A User’s Manual (Prentice-Hall Inc., Englewood Cliffs, NJ, 1990).Google Scholar
  58. 58.
    A. M. Arthurs and A. Dalgarno, Proc. R. Soc. London Ser. A 256, 540 (1960).MathSciNetADSzbMATHCrossRefGoogle Scholar
  59. 59.
    W. K. Trail, M. A. Morrison, W. A. Isaacs, and B. C. Saha, Phys. Rev. A 41, 4868 (1990).ADSCrossRefGoogle Scholar
  60. 60.
    R. G. Newton, Scattering Theory of Waves and Particles (Second Edition), (New York: Springer-Verlag, 1982).zbMATHGoogle Scholar
  61. 61.
    S. Geltman, Topics in Atomic Collision Theory (New York: Academic Press, 1969).Google Scholar
  62. 62.
    L. S. Rodberg and R. M. Thaler, Introduction to the Quantum Theory of Scattering (New York: Academic Press, 1967).Google Scholar
  63. 63.
    M. A. Morrison, N. F. Lane, and L. A. Collins, Phys. Rev. A 15, 2186 (1977).ADSCrossRefGoogle Scholar
  64. 64.
    D. T. Birtwistle and A. Herzenberg, J. Phys. B 4, 53 (1971).ADSCrossRefGoogle Scholar
  65. 65.
    B. K. Elza, Ph. D. thesis, University of Oklahoma, 1992.Google Scholar
  66. 66.
    R. K. Nesbet, Variational Methods in Electron-Atom Scattering Theory (New York: Plenum, 1980).CrossRefGoogle Scholar
  67. 67.
    U. Fano, Comments At. Mol. Phys. 1, 140 (1970).Google Scholar
  68. 68.
    E. S. Chang and U. Fano, Phys. Rev. A 6, 173 (1972).ADSCrossRefGoogle Scholar
  69. 69.
    K. A. Jerjian and R. J. W. Henry, Phys. Rev. A 31, 585 (1985).ADSCrossRefGoogle Scholar
  70. 70.
    M. A. Morrison, B. C. Saha, and A. N. Feldt, Phys. Rev. A 30, 2811 (1984).ADSCrossRefGoogle Scholar
  71. 71.
    A. Temkin and F. H. M. Faisal, Phys. Rev. A 3, 520 (1971).ADSCrossRefGoogle Scholar
  72. 72.
    M. Shugard, and A. Hazi, Phys. Rev. A 12, 1895 (1975).ADSCrossRefGoogle Scholar
  73. 73.
    M. A. Morrison, M. Abdolsalami, and B. K. Elza, Phys. Rev. A 43, 3440 (1991).ADSCrossRefGoogle Scholar
  74. 74.
    A. C. Allison, Adv. At. Mol. Phys. 25, 323 (1988).ADSCrossRefGoogle Scholar
  75. 75.
    W. N. Sams and D. J. Kouri, J. Chem. Phys. 51, 4809 (1969).MathSciNetADSCrossRefGoogle Scholar
  76. 76.
    N. F. Lane and S. Geltman, Phys. Rev. 160, 53 (1967).ADSCrossRefGoogle Scholar
  77. 77.
    T. N. Rescigno and A. E. Orel, Phys. Rev. A 25, 2402 (1982).ADSCrossRefGoogle Scholar
  78. 78.
    M. A. Morrison, Comput. Phys. Commun. 21, 63 (1980).ADSCrossRefGoogle Scholar
  79. 79.
    L. A. Collins, D. W. Norcross, and G. B. Schmid, Comput. Phys. Commun. 79, 63 (1980).Google Scholar
  80. 80.
    M. A. Morrison and L. A. Collins, J. Phys. B 10, L119 (1977).ADSCrossRefGoogle Scholar
  81. 81.
    W. M. Huo, T. L. Gibson, M. A. P. Lima, and V. McKoy, Phys. Rev. A 36, 1632 (1987).ADSCrossRefGoogle Scholar
  82. 82.
    H.-D. Meyer, Phys. Rev. A 40, 5605 (1989).ADSCrossRefGoogle Scholar
  83. 83.
    L. Castillejo, I. C. Percival, and M. J. Seaton, Proc. R. Soc. London, Ser. A 254, 259 (1960).MathSciNetADSzbMATHCrossRefGoogle Scholar
  84. 84.
    M. A. Morrison and P. J. Hay, Phys. Rev. A 20, 740 (1979).ADSCrossRefGoogle Scholar
  85. 85.
    C. A. Weatherford, K. Onda, and A. Temkin, J. Phys. B 31, 3620 (1985).Google Scholar
  86. 86.
    B. K. Elza, T. L. Gibson, M. A. Morrison, and B. C. Saha, J. Phys. B 22, 113 (1989).ADSCrossRefGoogle Scholar
  87. 87.
    I. C. Percival and M. J. Seaton, Prof. Cambridge Phil. Soc. 53, 654 (1957).ADSzbMATHCrossRefGoogle Scholar
  88. 88.
    H. S. W. Massey, and I. C. Percival, Proc. Roy. Soc. A 274, 427 (1963).ADSCrossRefGoogle Scholar
  89. 89.
    G. Herzberg, Molecular Spectra and Molecular Structure I: Spectra of Diatomic Molecules (Second Edition) (Van Nostrand, New York, 1950).Google Scholar
  90. 90.
    K. P. Huber, and G. Herzberg, Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules (Van Nostrand, New York, 1979).Google Scholar
  91. 91.
    A. A. Radzig and B. Smirnov, Reference Data on Atoms, Molecules, and Ions (Springer-Verlag, New York, 1986).Google Scholar
  92. 92.
    P. M. Morse, Phys. Rev. 34, 57 (1929).ADSzbMATHCrossRefGoogle Scholar
  93. 93.
    M. Abramowitz and I. A. Stegun, Pocketbook of Mathematical Functions (Frankfurt: Deutsch, 1984).zbMATHGoogle Scholar
  94. 94.
    J. Spanier and K. B. Oldham, An Atlas of Functions (New York: Hemisphere, 1987).zbMATHGoogle Scholar
  95. 95.
    G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Second Edition), (New York: McGraw Hill, 1968).Google Scholar
  96. 96.
    S. Wolfram, Mathematica: A System for Doing Mathematics by Computer (Second Edition) (New York: Adison-Wesley, 1991).Google Scholar
  97. 97.
    Y. P. Varshni, Rev. Mod. Phys. 29, 664 (1957).ADSCrossRefGoogle Scholar
  98. 98.
    D. Steele, E. R. Lippincott, and J. T. Vanderslice, Rev. Mod. Phys. 34, 239 (1962).ADSCrossRefGoogle Scholar
  99. 99.
    G. Simons, R. G. Parr, and J. M. Finlan, J. Chem. Phys. 59, 3229 (1973).ADSCrossRefGoogle Scholar
  100. 100.
    J. M. Finlan and G. Simons, J. Mol. Spectrosc. 57, 1 (1975).ADSCrossRefGoogle Scholar
  101. 101.
    D. W. Norcross and N. T. Padial, Phys. Rev. A 25, 226 (1982).ADSCrossRefGoogle Scholar
  102. 102.
    L. A. Collins and D. W. Norcross Phys. Rev. A 18, 467 (1978).ADSCrossRefGoogle Scholar
  103. 103.
    N. Chandra, Phys. Rev. A 16, 80 (1977).ADSCrossRefGoogle Scholar
  104. 104.
    E. Gerjuoy, and S. Stein Phys. Rev. 97, 1671 (1955).ADSCrossRefGoogle Scholar
  105. 105.
    A. Dalgarno and R. J. Moffett, Proc. Natl. Acad. Sci. India 33, 511 (1963).Google Scholar
  106. 106.
    L. G. H. Huxley and R. W. Crompton, The Diffusion and Drift of Electrons in Gases (Wiley: New York, 1974).Google Scholar
  107. 107.
    U. Fano and D. Dill, Phys. Rev. A 6, 185 (1972).ADSCrossRefGoogle Scholar
  108. 108.
    W. A. Isaacs and M. A. Morrison, J. Phys. B 25, 703 (1992).ADSCrossRefGoogle Scholar
  109. 109.
    I. I. Fabrikant, J. Phys. B 17, 4223 (1984).ADSCrossRefGoogle Scholar
  110. 110.
    M. A. Morrison, R. W. Crompton, B. C. Saha, and Z. LjPetrovic, Aust. J. Phys. 40, 239 (1987).ADSCrossRefGoogle Scholar
  111. 111.
    A. U. Hazi, Phys. Rev. A 19, 920 (1979).ADSCrossRefGoogle Scholar
  112. 112.
    C. W. Clark, Phys. Rev. A 30, 750 (1984).ADSCrossRefGoogle Scholar
  113. 113.
    G. Forsythe and C. B. Moler, Computational Solutions of Linear Algebraic Systems (Englewood Cliffs, NJ: Prentice-Hall, 1967).Google Scholar
  114. 114.
    M. A. Morrison, and N. F. Lane, Chem. Phys. Lett. 66, 527 (1979).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Michael A. Morrison
    • 1
  • Weiguo Sun
    • 1
  1. 1.Department of Physics and AstronomyUniversity of OklahomaNormanUSA

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