The Self-Consistent Field Equations for Generalized Valence Bond and Open-Shell Hartree—Fock Wave Functions

  • Frank W. Bobrowicz
  • William A. GoddardIII
Part of the Modern Theoretical Chemistry book series (MTC, volume 3)

Abstract

The basic starting point for calculating ab initiowave functions of molecules is generally the Hartree—Fock (HF) wave function, which in the simplest case involves two electrons (one with each spin) in each orbital Φi with the total wave function antisymmetrized in order to satisfy the Pauli principle
$$ a[({\phi _{1}}\alpha )({\phi _{1}}\beta )({\phi _{2}}\alpha )({\phi _{2}}\beta )...({\phi _{n}}\alpha )({\phi _{n}}\beta )] = a[({\phi _{1}}{\phi _{1}}{\phi _{2}}{\phi _{2}}...{\phi _{n}}{\phi _{n}}\alpha \beta \alpha \beta ...\alpha \beta )] $$
(1)
Here a is the antisymmetrizer or determinant operator* and α and β are the usual spin functions. In Eq. (1) as elsewhere, we arrange products of spatial functions and spin functions in order of increasing electron numbers.

Keywords

Wave Function Energy Expression Occupied Orbital Optimum Orbital Generalize Valence 
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.
    W. A. Goddard III and R. C. Ladner, A generalized orbital description of the reactions of small molecules, J. Am. Chem. Soc. 93, 6750–6756 (1971).CrossRefGoogle Scholar
  2. 2.
    R. C. Ladner and W. A. Goddard III, Improved quantum theory of many-electron systems. V. The spin-coupling and optimized GI method, J. Chem. Phys. 51, 1073–1087 (1969).CrossRefGoogle Scholar
  3. 3.
    W. J. Hunt, P. J. Hay, and W. A. Goddard III, Self-consistent procedures for generalized valence bond wave functions. Applications H3, BH, H2O, C2H6, and O2, J. Chem. Phys. 57, 738–748 (1972).CrossRefGoogle Scholar
  4. 4.
    C. C. J. Roothaan, Self-consistent field theory for open shells of electronic systems, Rev. Mod. Phys. 32, 179–185 (1960).CrossRefGoogle Scholar
  5. 5.
    F. W. Bobrowicz, Ph.D. thesis, California Institute of Technology (1974).Google Scholar
  6. 6.
    W. J. Hunt, W. A. Goddard III, and T. H. Dunning Jr., The incorporation of quadratic convergence into open-shell self-consistent field equations, Chem. Phys. Lett. 6, 147–151 (1970).CrossRefGoogle Scholar
  7. 7.
    W. A. Goddard III, Improved Quantum Ttheory of many-electron systems. II. The basic method, Phys. Rev. 157, 81–93 (1967).CrossRefGoogle Scholar
  8. 8.
    B. J. Moss and W. A. Goddard III, Configuration interaction studies on low-lying states of O2, J. Chem. Phys. 63, 3523–3531 (1975).CrossRefGoogle Scholar
  9. 9.
    T. Arai, Theorem on separability of electron pairs, J. Chem. Phys. 33, 95–98 (1960).CrossRefGoogle Scholar
  10. 10.
    P.-O. Löwdin, Note on the separability theorem for electron pairs, J. Chem. Phys. 35, 78–81 (1961).CrossRefGoogle Scholar
  11. 11.
    G. Levin and W. A. Goddard III, The generalized valence bond description of allyl radical, J. Am. Chem. Soc. 97, 1649–1656 (1975).CrossRefGoogle Scholar
  12. 12.
    G. Levin, Ph.D. thesis, California Institute of Technology, April 1974.Google Scholar
  13. 13.
    B. J. Moss, F. W. Bobrowicz, and W. A. Goddard III, The generalized valence bond description of O2, J. Chem. Phys. 63, 4632–4639 (1975).CrossRefGoogle Scholar
  14. 14.
    P.-O. Löwdin and H. Shull, Natural orbitals in the quantum theory of two-electron systems, Phys. Rev. 101, 1730–1739 (1956).CrossRefGoogle Scholar
  15. 15.
    J. M. Parks and R. G. Parr, Theory of separated electron pairs, J. Chem. Phys. 28, 335–345 (1958).CrossRefGoogle Scholar
  16. 16.
    D. M. Silver, E. L. Mehler, and K. Ruedenberg, Electron correlation and separated pair approximation in diatomic molecules. I. Theory, J. Chem Phys. 52, 1174–1180 (1970).CrossRefGoogle Scholar
  17. 17.
    C. C. J. Roothaan, New developments in molecular orbital theory, Rev. Mod. Phys. 23, 69–89 (1951).CrossRefGoogle Scholar
  18. 18.
    I. Shavitt, C. F. Bender, A. Pipano, and R. P. Hosteny, The iterative calculation of several of the lowest or highest eigenvalues and corresponding eigenvalues of very large symmetric matrices, J. Comput. Phys., 11 90–108 (1973).CrossRefGoogle Scholar
  19. 19.
    R. C. Raffenetti, Preprocessing two-electron integrals for efficient utilization in many-electron self-consistent field calculations, Chem. Phys. Lett. 20, 335–338 (1973).CrossRefGoogle Scholar
  20. 20.
    R. K. Nesbet, Configuration interaction in orbital systems, Proc. Roy. Soc. London, Ser. A 230, 312–321 (1955).CrossRefGoogle Scholar
  21. 21.
    F. W. Birss and S. Fraga, Self-consistent-field theory. I. General Treatment, J. Chem. Phys. 38, 2552–2557 (1963).CrossRefGoogle Scholar
  22. 22.
    G. Das, Extended Hartree-Fock ground-state wavefunctions for the lithium molecule, J. Chem. Phys. 46, 1568–1579 (1967).CrossRefGoogle Scholar
  23. 23.
    B. Levy, Best choice for the coupling operators in the open-shell and multiconfiguration SCF methods, J. Chem. Phys. 48, 1994–1996 (1968).CrossRefGoogle Scholar
  24. 24.
    J. Hinze and C. C. J. Roothaan, Multiconfiguration self-consistent field theory, Prog. Theor. Phys. (Kyoto) Supp. 40, 37–51 (1967).CrossRefGoogle Scholar
  25. 25.
    B. Levy, Best choice for the coupling operators in the open-shell and multiconfiguration SCF methods, J. Chem. Phys. 48, 1994–1996 (1968).CrossRefGoogle Scholar
  26. 26.
    S. Huzinaga, Coupling operator method in the SCF theory, J. Chem. Phys. 51, 3971–3975 (1969).CrossRefGoogle Scholar
  27. 27.
    W. J. Hunt, T. H. Dunning Jr., and W. A. Goddard III, 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
  28. 28.
    S. Huzinaga, Analytical methods in Hartree-Fock self-consistent field theory, Phys. Rev., 122 131–138 (1961).CrossRefGoogle Scholar
  29. 29.
    D. Peters, Simple open-shell SCF molecular orbital computations, J. Chem. Phys., 57 4351–4353 (1972).CrossRefGoogle Scholar
  30. 30.
    W. A. Goddard III, T. H. Dunning Jr., and W. J. Hunt, The proper treatment of off-diagonal Lagrange multipliers and coupling operators in self-consistent field equations, Chem. Phys. Lett., 4 231–234 (1969).CrossRefGoogle Scholar
  31. 31.
    J. P. Dahl, H. Johansen, D. R. Truax, and T. Ziegler, On the derivation of necessary conditions on Hartree-Fock orbitals, Chem. Phys. Lett., 6 64–66 (1970).CrossRefGoogle Scholar
  32. 32.
    R. Albat and N. Gruen, Examples of known SCF procedures which do not satisfy all necessary conditions for the energy to be stationary, Chem. Phys. Lett., 18 572–573 (1973).CrossRefGoogle Scholar
  33. 33.
    K. Hirao and H. Nakatsuji, General SCF operator satisfying correct variational condition, J. Chem. Phys., 59 1457–1462 (1973).CrossRefGoogle Scholar
  34. 34.
    E. R. Davidson, Spin-restricted open-shell self-consistent-field theory, Chem. Phys. Lett., 21 565–567 (1973).CrossRefGoogle Scholar
  35. 35.
    M. H. Wood and A. Veillard, On convergence guarantees for the multiconfiguration selfconditions for the energy to be stationary, Chem. Phys. Lett., 18 572–573 (1973).CrossRefGoogle Scholar
  36. 36.
    M. Rossi, Variational procedure for open-shell LCAO multideterminant wavefunctions. An approach to the excited-state problem, J. Chem. Phys., 46 989–996 (1967).CrossRefGoogle Scholar
  37. 37.
    N. G. Mukherjee, A variational procedure for the optimization of multi-configuration self-consistent field (MCSCF) orbitals, Chem. Phys. Lett., 24 441–446 (1974).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1977

Authors and Affiliations

  • Frank W. Bobrowicz
    • 1
    • 2
  • William A. GoddardIII
    • 1
  1. 1.Arthur Amos Noyes Laboratory of Chemical PhysicsCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Batelle Memorial InstituteColumbusUSA

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