The Equations of Motion Method: An Approach to the Dynamical Properties of Atoms and Molecules

  • Clyde W. McCurdyJr.
  • Thomas N. Rescigno
  • Danny L. Yeager
  • Vincent McKoy
Part of the Modern Theoretical Chemistry book series (MTC, volume 3)


This chapter is concerned with the equations of motion method as a many-body approach to the dynamical properties of atoms and molecules. In a wide range of spectroscopic experiments one is primarily concerned with just dynamical properties. These dynamical properties include excitation energies and oscillator strengths in optical spectroscopy, the dynamic or frequency-dependent polarizability in light scattering studies, photoionization cross sections, and elastic and inelastic electron scattering cross sections. These experiments probe the response of an atom or molecule to some external perturbation. If one is concerned with these properties one should develop a formalism which aims directly at these properties. Of course this idea is not novel. For example, one might try to calculate the appropriate Green’s functions whose poles, and residues at these poles, are directly the excitation energies and transitions densities, respectively. One could also attempt to solve the time-dependent Schrödinger equation directly, e.g., in the time-dependent Hartree—Fock approximation. The approach to these dynamical properties of atoms and molecules which we will discuss is based on the equations of motion formalism as suggested by Rowe.(1) This is a very practical formalism based on the equations of motion for excitation operators defined as operators that convert one stationary state of a system into another state.


Excitation Energy Oscillator Strength Electron Affinity Internuclear Distance Potential Energy Curve 
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  1. 1.
    D. J. Rowe, Equations-of-motion method and the extended shell model, Rev. Mod. Phys. 40, 153–166 (1968).CrossRefGoogle Scholar
  2. 2.
    T. Shibuya and V. McKoy, Higher random phase approximation as an approximation to the equations of motion, Phys. Rev. A 2, 2208–2218 (1970).CrossRefGoogle Scholar
  3. 3.
    T. Shibuya, J. Rose, and V. McKoy, Equations-of-motion method including renormalization and double-excitation mixing, J. Chem. Phys. 58, 500–507 (1973).CrossRefGoogle Scholar
  4. 4.
    J. Rose, T. Shibuya, and V. McKoy, Application of the equations-of-motion method to the excited states of N2, CO, and C2H4, J. Chem. Phys. 58, 74–83 (1973).CrossRefGoogle Scholar
  5. 5.
    C. W. McCurdy, Jr. and V. McKoy, Equations of motion method: Inelastic electron scattering for helium and CO2 in the Born approximation, J. Chem. Phys. 61, 2820–2826 (1974).CrossRefGoogle Scholar
  6. 6.
    D. L. Yeager and V. McKoy, Equations of motion method: Excitation energies and intensities of formaldehyde, J. Chem. Phys. 60, 2714–2716 (1974).CrossRefGoogle Scholar
  7. 7.
    J. Rose, T. Shibuya, and V. McKoy, Electronic excitations of benzene from the equations of motion method, J. Chem. Phys. 60, 2700–2702 (1974).CrossRefGoogle Scholar
  8. 8.
    D. J. Rowe, General variational equations for stationary and time-dependent states, Nucl. Phys. A 107, 99–105 (1968).CrossRefGoogle Scholar
  9. 9.
    D. J. Rowe, Nuclear Collective Motion, Models, and Theory, Methuen and Co. Ltd., London (1970).Google Scholar
  10. 10.
    See, for example, A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York (1971).Google Scholar
  11. 11.
    P. H. S. Martin, W. H. Henneker, and V. McKoy, Dipole properties of atoms and molecules in the random phase approximation, J. Chem. Phys. 62, 69–79 (1975).CrossRefGoogle Scholar
  12. 12.
    D. J. Thouless, Vibrational states of nuclei in the random phase approximation, Nucl. Phys. 22, 78–95 (1961).CrossRefGoogle Scholar
  13. 13.
    D. L. Yeager and V. McKoy, An equations of motion approach for open shell systems, J. Chem. Phys. 63, 4861 (1975).CrossRefGoogle Scholar
  14. 14.
    W. J. Hunt, T. H. Dunning Jr., , and W. A. Goddard, The orthogonality constrained basis set expansion method for treating off-diagonal Lagrange multipliers in calculations of electronic wave functions, Chem. Phys. Lett. 3, 606–610 (1969).CrossRefGoogle Scholar
  15. 15.
    P. Jorgensen, Electronic excitations of open-shell systems in the grand canonical and canonical time-dependent Hartree—Fock models. Applications on hydrocarbon radical ions, J. Chem. Phys. 57, 4884–4892 (1972).CrossRefGoogle Scholar
  16. 16.
    L. Armstrong Jr., An open-shell random phase approximation, J. Phys. B 7, 2320–2331 (1974).CrossRefGoogle Scholar
  17. 17.
    W. Coughran, J. Rose, T. Shibuya, and V. McKoy, Equations of motion method: Potential energy curves for N2, CO, and C2H4, J. Chem. Phys. 58, 2699–2709 (1973).CrossRefGoogle Scholar
  18. 18.
    K. Dressler, The lowest valence and Rydberg states in the dipole-allowed absorption spectrum of nitrogen. A survey of their interactions. Can. J. Phys. 47, 547–561 (1969).CrossRefGoogle Scholar
  19. 19.
    H. Lefebvre-Brion, Theoretical study of homogeneous perturbations. II. Least-squares fitting method to obtain “deperturbed” crossing Morse curves. Application to the perturbed 1Σ states of N2, Can. J. Phys. 47, 541–546 (1969).CrossRefGoogle Scholar
  20. 20.
    E. Lassettre and A. Skerbele, Absolute generalized oscillator strengths for four electronic transitions in carbon monoxide, J. Chem. Phys. 54, 1597–1607 (1971).CrossRefGoogle Scholar
  21. 21.
    K. N. Klump and E. N. Lassettre, Relative vibrational intensities for the B 1 Σ + F-X 1Σ+ transition in carbon monoxide, J. Chem. Phys. 60, 4830–4832 (1974).CrossRefGoogle Scholar
  22. 22.
    The basis set used in these calculations is different from that of Ref. 4. See Ref. 15 for details.Google Scholar
  23. 23.
    G. Herzberg, T. Hugo, S. Tilford, and J. Simmons, Rotational analysis of the forbidden d 3 i F-X 1Σ+ absorption bands of carbon monoxide, Can. J. Phys. 48, 3004–3015 (1970).CrossRefGoogle Scholar
  24. 24.
    P. H. Krupenie and S. Weiss, Potential energy curves for CO and CO+, J. Chem. Phys. 43, 1529–1534 (1965).CrossRefGoogle Scholar
  25. 25.
    V. D. Meyer, A. Skerbele, and E. N. Lassettre, Intensity distribution in the electron-impact spectrum of carbon monoxide at high-resolution and small scattering angles, J. Chem. Phys. 43, 805–816 (1965).CrossRefGoogle Scholar
  26. 26.
    M. J. Mumma, E. J. Stone, and E. C. Zipf, Excitation of the CO fourth positive band system by electron impact on carbon monoxide and carbon dioxide, J. Chem. Phys. 54, 2627–2634 (1971).CrossRefGoogle Scholar
  27. 27.
    P. G. Wilkinson, Absorption spectra of ethylene and ethylene-d4 in the vacuum ultraviolet. II. Can. J. Phys. 34, 643–652 (1956).CrossRefGoogle Scholar
  28. 28.
    C. F. Bender, T. H. Dunning Jr., , H. F. Schaefer III, W. A. Goddard III, and W. J. Hunt, Multiconfiguration wavefunctions for the lowest (iπππ*) excited states of ethylene, Chem. Phys. Lett. 15, 171–178 (1972).CrossRefGoogle Scholar
  29. 29.
    R. J. Buenker and S. D. Peyerimhofif, All-valence-electron CM calculations for the characterization of the 1(iπ, iT*) states of ethylene, in press.Google Scholar
  30. 30.
    M. Inokuti, Inelastic collisions of fast charged particles with atoms and molecules. The Bethe theory revisited, Rev. Mod. Phys. 43, 297–347 (1971).CrossRefGoogle Scholar
  31. 31.
    E. N. Lassettre and J. C. Shiloff, Collision cross-section study of CO2, J. Chem. Phys. 43, 560–571 (1965).CrossRefGoogle Scholar
  32. 32.
    J. W. Rabalais, J. M. McDonald, V. Scherr, and S. P. McGlynn, Electronic spectroscopy of isoelectronic molecules. II. Linear triatomic groupings containing sixteen valence electrons, Chem. Rev. 71, 73–108 (1971).CrossRefGoogle Scholar
  33. 33.
    For the results of extensive CI calculations, see N. W. Winter, C. F. Bender, and W. A. Goddard III, Theoretical assignments of the low-lying electronic states of carbon dioxide, Chem. Phys. Lett. 20, 489–492 (1973).CrossRefGoogle Scholar
  34. 34.
    M. Krauss, S. R. Mielczarek, D. Neumann, and C. E. Kuyatt, Mechanism for production of the fourth positive band system of CO by electron impact on CO2, J. Geophys. Res. 76, 3733–3737 (1971).CrossRefGoogle Scholar
  35. 35.
    G. M. Lawrence, Photodissociation of CO2 to produce CO(a3II), J. Chem. Phys. 56, 3435–3442 (1972).CrossRefGoogle Scholar
  36. 36.
    V. J. Hammond and W. C. Price, Oscillator strengths of the vacuum ultraviolet absorption bands of benzene and ethylene, Trans. Faraday Soc. 51, 605–610 (1955).CrossRefGoogle Scholar
  37. 37.
    See, for example, E. Clementi and A. D. McLean, Atomic negative ions, Phys. Rev. 133, A419-A423 (1964).CrossRefGoogle Scholar
  38. 38.
    D. L. Yeager, Ph.D. candidacy examination report, California Institute of Technology, March 1972.Google Scholar
  39. 39.
    J. Simons and W. D. Smith, Theory of electron affinities of small molecules, J. Chem. Phys. 58, 4899–4907 (1973).CrossRefGoogle Scholar
  40. 40.
    L. S. Cederbaum, G. Hohlneicher, and W. V. Niessen, Improved calculations of ionization potentials of closed-shell molecules, Mol. Phys. 26, 1405–1424 (1973).CrossRefGoogle Scholar
  41. 41.
    G. D. Purvis and Y. Ohπn, Atomic and molecular electronic spectra and properties from the electron propagator, J. Chem. Phys. 60, 4063–4069 (1974).CrossRefGoogle Scholar
  42. 42.
    D. L. Yeager, Ph.D. thesis, California Institute of Technology (February 1975).Google Scholar
  43. 43.
    T. Chen, W. Smith and J. Simons, Theoretical studies of molecular ions. Vertical ionization potentials of the nitrogen molecule, Chem. Phys. Lett. 26, 296–300 (1974).CrossRefGoogle Scholar
  44. 44.
    W. Smith, T. Chen, and J. Simons, Theoretical studies of molecular ions. Vertical ionization potentials of hydrogen fluoride, J. Chem. Phys. 61, 2670–2674 (1974).CrossRefGoogle Scholar
  45. 45.
    W. D. Smith, T. Chen, and J. Simons, Theoretical studies of molecular ions. Vertical detachment energy of OH-, Chem. Phys. Lett. 27, 499–502 (1974).CrossRefGoogle Scholar
  46. 46.
    J. T. Broad and W. P. Reinhardt,Calculation of photoionization cross sections using L2 basis sets, J. Chem. Phys. 60, 2182–2183 (1974).CrossRefGoogle Scholar
  47. 47.
    T. N. Rescigno, C. W. McCurdy, and V. McKoy, Calculation of helium photoionization in the random phase approximation using square-integrable basis functions, Phys. Rev. A 9, 2409–2412 (1974).CrossRefGoogle Scholar
  48. 48.
    P. H. S. Martin, T. N. Rescigno, V. McKoy, and W. H. Henneker, Photoionization cross sections for H2 in the random phase approximation with a square-integrable basis, Chem. Phys. Lett. 29, 496–501 (1974).CrossRefGoogle Scholar
  49. 49.
    P. W. Langhoff, Stieltjes imaging of atomic and molecular photoabsorption profiles, Chem. Phys. lett. 22, 60–64 (1973).CrossRefGoogle Scholar
  50. 50.
    P. W. Langhoff and C. T. Corcoran, Stieltjes imaging of photoabsorption and dispersion profiles, J. Chem. Phys. 61, 146–159 (1974).CrossRefGoogle Scholar
  51. 51.
    A. Dalgarno, H. Doyle, and M. Oppenheimer, Calculation of photoabsorption processes in helium, Phys. Rev. Lett. 29, 1051–1052 (1972).CrossRefGoogle Scholar
  52. 52.
    H. Doyle, M. Oppenheimer, and A. Dalgarno, Bound-state expansion method for calculating resonance and nonresonance contributions to continuum processes: Theoretical development and application to the photoionization of helium, Phys Rev. A 11, 909 (1975).CrossRefGoogle Scholar
  53. 53.
    M. Ya. Amus’ya, N. A. Cherepkov, and L. V. Chernysheva, Cross sections for the photo-ionization of noble-gas atoms with allowance for multielectron correlations, Soy. Phys.-JETP 33, 90–96 (1971).Google Scholar
  54. 54.
    M. J. Jamieson, Time-dependent Hartree-Fock theory for atoms, Int. J. Quantum Chem. S4, 103–115 (1971).Google Scholar
  55. 55.
    P. L. Altick and A. E. Glassgold, Correlation effects in atomic structure using the random phase approximation, Phys. Rev. 133, A632-A646 (1964).CrossRefGoogle Scholar
  56. 56.
    P. W. Langhoff and M. Karplus, Padé approximants to the normal dispersion expansion of dynamic polarizabilities, J. Chem. Phys. 52, 1435–1449 (1970).CrossRefGoogle Scholar
  57. 57.
    U. Fano and J. W. Cooper, Spectral distribution of atomic oscillator strengths, Rev. Mod. Phys. 40, 441–507 (1968).CrossRefGoogle Scholar
  58. 58.
    The quadrature-like approximation implicit in the use of an L 2 -basis set has been examined in the context of Fredholm scattering calculations. See E. J. Heller, T. N. Rescigno, and W. P. Reinhardt, Extraction of accurate scattering information from Fredholm determinants calculated in an L2 basis: A Chebyschev discretization of the continuum, Phys. Rev. A 8, 2946–2951 (1973).CrossRefGoogle Scholar
  59. 59.
    L. Schlessinger and C. Schwartz, Analyticity as a useful computational tool, Phys. Rev. Lett. 16, 1173–1174 (1966).CrossRefGoogle Scholar
  60. 60.
    L. Schlessinger, Use of analyticity in the calculation of nonrelativistic scattering amplitudes, Phys. Rev. 167, 1411–1423 (1968).CrossRefGoogle Scholar
  61. 61.
    H. S. Wall, The Analytic Theory of Continued Fractions, Van Nostrand, Princeton, New Jersey (1968).Google Scholar
  62. 62.
    D. L. Yeager, M. Nascimento, and V. McKoy, Some applications of excited state-excited state transition densities, Phys. Rev. A11, 1168 (1975).CrossRefGoogle Scholar
  63. 63.
    Y. M. Chan and A. Dalgarno, The dipole spectrum and properties of helium, Proc. Phys. Soc. London 86, 777–782 (1965).CrossRefGoogle Scholar
  64. 64.
    J. A. R. Samson, The measurement of the photoionization cross sections of the atomic gases, in: Advances in Atomic and Molecular Physics, Vol. 2, pp. 177–261, Academic Press, New York (1966).Google Scholar
  65. 65.
    D. W. Norcross, Photoionization of the metastable states, J. Phys. B 4, 652–657 (1971).CrossRefGoogle Scholar
  66. 66.
    V. L. Jacobs, Photoionization from excited states of helium, Phys. Rev. A 9, 1938–1946 (1974).CrossRefGoogle Scholar
  67. 67.
    R. F. Stebbings, F. B. Dunning, F. K. Tittel, and R. D. Rundel, Photoionization of helium metastable atoms near threshold, Phys. Rev. Lett. 30, 815–817 (1973).CrossRefGoogle Scholar
  68. 68.
    P. H. S. Martin, W. H. Henneker, and V. McKoy, Second-order optical properties and Van der Waals coefficients of atoms and molecules in the random phase approximation, Chem. Phys. Lett. 27, 52–56 (1974).CrossRefGoogle Scholar
  69. 69.
    A. L. Ford and J. C. Broione, Direct-resolvent-operator computations on the hydrogen’molecule dynamic polarizability, Rayleigh, and Raman scattering, Phys. Rev. A7, 418–426 (1973).CrossRefGoogle Scholar
  70. 70.
    L. Wolniewicz, Theoretical investigation of the transition probabilities in the hydrogen molecule, J. Chem. Phys. 51, 5002–5008 (1969).CrossRefGoogle Scholar
  71. 71.
    G. R. Cook and P. H. Metzger, Photoionization and absorption cross sections of H2 and D, in the vacuum ultraviolet region, J. Opt. Soc. Am. 54, 968–972 (1964).CrossRefGoogle Scholar
  72. 72.
    J. A. R. Samson and R. B. Cairns, Total absorption cross sections of H2, N2, and 02 in the region 550–200 A, J. Opt. Soc. Am. 55, 1035 (1965).Google Scholar
  73. 73.
    R. E. Rebbert and P. Ausloos, Ionization quantum yields and absorption coefifiicients of selected compounds at 58.4 and 73.6–74.4 nm, J. Res. Nat. Bur. Stand. Sect A. 75A, 481–485 (1971).Google Scholar
  74. 74.
    H. P. Kelly, The photoionization cross section for H2 from threshold to 30 eV, Chem. Phys. Lett. 20, 547–550 (1973).CrossRefGoogle Scholar
  75. 75.
    See P. G. Burke and M. J. Seaton, Numerical solutions of the integro-differential equations of electron—atom collision theory, in : Methods of Computational Physics (B. Alder, S. Fernbach, and M. Rotenberg, eds.), Vol. 10, pp. 1–80, Academic Press, New York (1971).Google Scholar
  76. 76.
    A. L. Fetter and K. M. Watson, The optical model, in : Advances in Theoretical Physics (K. Brueckner, ed.), Vol. 1, pp. 115–194, Academic Press, New York (1965).Google Scholar
  77. 77.
    H. Feshbach, A unified theory of nuclear reactions, Ann. Phys. (N. Y.) 5, 357–390 (1958);CrossRefGoogle Scholar
  78. 77.
    H. Feshbach, A unified theory of nuclear reactions. II, Ann. Phys. (N.Y.) 14, 287–313 (1962).CrossRefGoogle Scholar
  79. 78.
    J. S. Bell and E. J. Squires, A formal optical model, Phys. Rev. Lett. 3, 96–97 (1959).CrossRefGoogle Scholar
  80. 79.
    B. Schneider, H. S. Taylor, and R. Yaris, Many-body theory of the elastic scattering of electrons from atoms and molecules, Phys. Rev. A 1, 855–867 (1970).CrossRefGoogle Scholar
  81. 80.
    B. S. Yarlagadda, Gy. Csanak, H. S. Taylor, B. Schneider, and R. Yaris, Application of many-body Green’s functions to the scattering and bound-state properties of helium, Phys. Rev. A 7, 146–154 (1973).CrossRefGoogle Scholar
  82. 81.
    C. W. McCurdy, T. N. Rescigno, and V. McKoy, A many-body treatment of Feshbach theory applied to electron—atom and electron—molecule collisions, Phys. Rev. A 12, 406 (1975).CrossRefGoogle Scholar
  83. 82.
    K. Dietrich and K. Hara, On the many-body theory of nuclear reactions, Nucl. Phys. A 111, 392–416 (1968).Google Scholar
  84. 83.
    T. N. Rescigno, C. W. McCurdy, and V. McKoy, Discrete basis set approach to nonspherical scattering, Chem. Phys. Len, 27, 401–404 (1974);Google Scholar
  85. 83.
    T. N. Rescigno, C. W. McCurdy, and V. McKoy, Discrete basis set approach to nonspherical scattering. II, Phys. Rev. A 10, 2240–2245 (1974).CrossRefGoogle Scholar
  86. 84.
    T. N. Rescigno, C. W. McCurdy, and V. McKoy, Low-energy e--H2 elastic scattering cross sections using discrete basis functions, Phys. Rev. A 11, 825–829 (1975).CrossRefGoogle Scholar
  87. 85.
    T. N. Rescigno, C. W. McCurdy, and V. McKoy, A relationship between the many-body theory of inelastic scattering and the distorted wave, J. Phys. B7, 2396–2402 (1974).CrossRefGoogle Scholar
  88. 86.
    See for example, J. R. Taylor, Scattering Theory, p. 720, J. Wiley and Sons, New York (1972).Google Scholar
  89. 87.
    Gy. Csanak, H. S. Taylor, and R. Yaris, Many-body methods applied to electron scattering from atoms and molecules. II. Inelastic processes, Phys. Rev. A 3, 1322–1328 (1971).CrossRefGoogle Scholar
  90. 88.
    L. D. Thomas, B. S. Yarlagadda, Gy. Csanak, and H. S. Taylor, Analytical and numerical procedures in the application of many-body Green’s function methods to electron—atom scattering problems, Comput. Phys. Comm. 6, 316–330 (1973)CrossRefGoogle Scholar
  91. 88a.
    L. D. Thomas, B. S. Yarlagadda, Gy. Csanak, and H. S. Taylor The application of first order many-body theory to the calculation of differential and integral cross sections for the electron impact excitation of the 21S, 21P, 23S, 23P states of helium, J. Phys. B 7, 1719–1733 (1974).CrossRefGoogle Scholar
  92. 89.
    T. N. Rescigno, C. W. McCurdy, and V. McKoy, Excitation of the b 3Σu state of H2 by low energy electron impact in the distorted wave approximation (in preparation).Google Scholar
  93. 90.
    J. C. Tully and R. S. Berry, Elastic scattering of low energy electrons by the hydrogen molecule, J. Chem. Phys.51, 2056–2075 (1969).CrossRefGoogle Scholar
  94. 91.
    B. Schneider, Inelastic scattering of high-energy electrons from atoms: The helium atom, Phys. Rev. A 2, 1873–1877 (1970).CrossRefGoogle Scholar
  95. 92.
    A. Szabo and N. S. Ostlund, Generalized oscillator strengths for the lowest H F- Σ transitions in CO and N2, Chem. Phys. Lett. 17, 163–166 (1972).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1977

Authors and Affiliations

  • Clyde W. McCurdyJr.
    • 1
  • Thomas N. Rescigno
    • 1
  • Danny L. Yeager
    • 1
  • Vincent McKoy
    • 1
  1. 1.Arthur Amos Noyes Laboratory of Chemical PhysicsCalifornia Institute of TechnologyPasadenaUSA

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