Partitioning Technique for Determinantal Equations

  • Frank Weinhold


The partitioning of large arrays into block components which could themselves be manipulated as algebraic entities is a technique which has been used with particular effectiveness by P.-O. Löwdin and his school. This “partitioning technique” underlies the Löwdin analysis /1/ of the relationship between perturbation and variational treatments of Schrödinger’s equation, and leads to the resolvent algebra, inner projections, and other important formal developments. In this brief note we describe how this technique can also simplify the treatment of determinantal equations, which permit a unified approach to certain problems of numerical approximation, interpolation, and quadratures which arise frequently in quantum chemistry, as well as to the determination of rigorous error bounds for the quality of approximate wavefunctions and the associated quantum-mechanical properties /2/.


Determinantal Equation Partitioning Technique Block Component Rigorous Bound Inhomogeneous Linear Equation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    P.-O. Löwdin, in, C.H. Wilcox (ed.), Perturbation Theroy and Its Applications in Quantum Mechanics (John Wiley, New York, 1966) pp. 255–294, and references therein.Google Scholar
  2. 2.
    F. Weinhold, Advan. Quantum Chem. 6, 299 (1972).ADSCrossRefGoogle Scholar
  3. 3.
    E.A. Hylleraas and B. Undheim, Z. Phys. 65, 759 (1930)ADSzbMATHCrossRefGoogle Scholar
  4. J.K.L. MacDonald, Phys. Rev. 43, 830 (1933).ADSCrossRefGoogle Scholar
  5. 4.
    F. Weinhold, J. Math. Phys. 11, 2127 (1970).MathSciNetADSCrossRefGoogle Scholar
  6. 5.
    P.S.C. Wang, J. Chem. Phys. 22, 4464 (1970)ADSCrossRefGoogle Scholar
  7. P.S.C. Wang, Chem. Phys. Letters 11, 318 (1971).ADSCrossRefGoogle Scholar
  8. 6.
    R. Blau, A.R.P. Rau, L. Spruch, Phys. Rev. A 8, 119 (1973)ADSCrossRefGoogle Scholar
  9. 7.
    M. Cohen and T. Feldmann, Can. J. Phys. 48, 1681 (1970)ADSCrossRefGoogle Scholar
  10. M. Cohen, T. Feldmann, and R.P. McEachran, J. Phys. B 2, 193 (1972)ADSCrossRefGoogle Scholar
  11. J.G. Leopold, J. Katriel, and M. Cohen, Chem. Phys. 3Google Scholar
  12. J.G. Leopold, M. Cohen, and. J. Katriel, J. Phys. B 8, 513 (1975).ADSCrossRefGoogle Scholar
  13. 8.
    See, e.g., P. Bonelli and G.F. Majorino, Nuovo Cim. B 62, 209 (1970)ADSCrossRefGoogle Scholar
  14. R.R. Merkel, J. Chem. Phys. 62, 3198 (1975).ADSCrossRefGoogle Scholar
  15. 9.
    F. Weinhold. Proc. Roy. Soc. (London) A327, 209 (1972)ADSCrossRefGoogle Scholar
  16. F. Weinhold. J. Chem. Phys. 59, 355 (1973)ADSCrossRefGoogle Scholar
  17. D.P. Shong and F. Weinhold, Can. J. Chem. 51, 260 (1973)CrossRefGoogle Scholar
  18. M.T. Anderson and F. Weinhold, Phys. Rev. A 2, 118 (1974)ADSCrossRefGoogle Scholar
  19. M.T. Anderson and F. Weinhold, Phys. Rev. A 11, 442 (1975).ADSCrossRefGoogle Scholar
  20. 10.
    J.S. Sims and R.C. Whitten, Phys. Rev. A 8, 2220 (1973)ADSCrossRefGoogle Scholar
  21. J.S. Sims and J.R. Rumble, Jr., Phys. Rev. A 8, 2231 (1973)ADSCrossRefGoogle Scholar
  22. J.S. Sims, S.A. Hagstrom, and J.R. Rumble, Jr. (to be published).Google Scholar
  23. 11.
    See, e.g., E.N. Svendsen, Chem. Phys. Letters 13,425 (1972), and Refs. 5–8.ADSCrossRefGoogle Scholar
  24. 12.
    F. Weinhold. J. Phys. A 1, 655 (1968).ADSCrossRefGoogle Scholar
  25. 13.
    F. Weinhold. J. Phys. B 2, 517 (1969)ADSCrossRefGoogle Scholar
  26. F. Weinhold. J Chem. Phys. 50, 4136 (1969).ADSCrossRefGoogle Scholar
  27. 14.
    P.-O. Löwdin, Phys. Rev. 139, A357 (1965)CrossRefGoogle Scholar
  28. P.-O. Löwdin, J. Chem. Phys. 13, S175 (1965).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1976

Authors and Affiliations

  • Frank Weinhold
    • 1
  1. 1.Department of ChemistryStanford UniversityStanfordUSA

Personalised recommendations