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Approximation Methods for Stationary States

  • Ralph E. Christoffersen
Part of the Springer Advanced Texts in Chemistry book series (SATC)

Abstract

In several of the preceding chapters we have discussed application of the postulates of quantum mechanics to several important types of examples. Although these applications were varied, there is one common characteristic shared by all of them. In each case, the Schrödinger equation could be solved exactly for the eigenvalues and associated eigenfunctions. If this were possible for all problems of interest to chemists, our insight into the nature of chemical phenomena would certainly be increased enormously. However, the fact is that only a very few problems are solvable exactly. In particular, it has not yet been possible to obtain an exact solution to the Schrödinger equation for any system containing more than one electron. Consequently, our knowledge of chemistry as revealed through quantum mechanics has been restricted primarily to that obtainable from an examination of approximate solutions to the Schrödinger equation. However, the advent of large-scale computers has greatly expanded both the accuracy and the scope of approximate solutions that can be obtained. We shall now discuss some of the methods that are available for obtaining approximate solutions to the Schrödinger equation for stationary states, and that form the basis for most applications of quantum mechanics to chemical problems.

Keywords

Perturbation Theory External Electric Field Helium Atom Trial Function Unperturbed System 
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Reference

  1. 1.
    This result was shown by C. E. Eckert, Phys. Rev., 36, 878 (1930).Google Scholar
  2. 1.
    H. Eyring, J. Walter, and G. E. Kimball, Quantum Chemistry, John Wiley, New York, 1944, pp. 103–106Google Scholar
  3. 9.
    E. A. Hylleraas and B. Undheim, Z. Phys., 65, 759 (1930); J. K. L. MacDonald, Phys. Rev., 43, 830 (1933); H. Shull and P. 0. Löwdin, Phys. Rev., 110, 1466 (1958).Google Scholar
  4. 12.
    See S. F. Boys, Proc. R. Soc. (London), A200 542 (1950) for the details of the integral evaluation. See also I. Shavitt, Meth. Computat. Phys., 2, 1 (1963).CrossRefGoogle Scholar
  5. 13.
    See J. V. L. Longstaff and K. Singer, Proc. R. Soc. (London) A258 421 (1960) for the original report and discussion of this example.Google Scholar
  6. 15.
    G. Temple, Proc. R. Soc. (London), A119, 276 (1928).CrossRefGoogle Scholar
  7. 16.
    The form presented in Eq. (9–96) was given by D. H. Weinstein, Phys. Rev., 40, 797; 41, 839 (1932); Proc. Nat. Acad. Sci. U.S.A., 20, 529 (1934), and J. K. L. MacDonald, Phys. Rev., 46, 828 (1934). See also A. F. Stevenson and M. F. Crawford, Phys. Rev., 54, 375 (1938). For a review of work in this area, see F. Weinhold, Adv. Quantum Chem., 6, 299–331 (1972).Google Scholar
  8. 17.
    I. T. Keaveny and R. E. Christoffersen, J. Chem. Phys., 50, 80–85 (1969).CrossRefGoogle Scholar
  9. 18.
    The “virial theorem” will be discussed in Chapter 10, Section 10–6. For its application to lower bounds, see G. L. Caldow and C. A. Coulson, Proc. Cambridge Philos. Soc.,57 341 (1961).Google Scholar
  10. 20.
    N. W. Bazley, Proc. Natl. Acad. Sci. U.S.A., 45, 850 (1959); Phys. Rev., 120, 44 (1960). See also N. W. Bazley and D. W. Fox, J. Res. Nat. Bus. Stand., B65, 105 (1961); Phys. Rev., 124, 483 (1961); J. Math. Phys., 3, 469 (1962).CrossRefGoogle Scholar
  11. 31.
    L. Brillouin, J. Phys. Radium, 3, 373 (1935); E. Wigner, Math. Naturw. Anz. ungar. Akad. Wiss., 53, 477 (1935). The presentation used here is based on the analysis using partitioning techniques of P. 0. Löwdin, J. Chem Phys., 19, 1396 (1951).Google Scholar
  12. 34.
    As an illustration of the application of BWPT-type techniques see, for example, G. A. Segal and R. W. Wetmore, Chem. Phys. Letters, 32, 556–560 (1975); J. J. Diamond, G. A. Segal and R. W. Wetmore, J. Phys. Chem., 88, 3532–3538 (1984); for alternative approaches, see R. J. Bartlett, J. C. Bellum, and E. J. Brändas, Int. J. Quantum Chem., 57, 449–462 (1973).Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • Ralph E. Christoffersen
    • 1
  1. 1.The Upjohn CompanyKalamazooUSA

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