Advertisement

Theoretica chimica acta

, Volume 44, Issue 1, pp 9–26 | Cite as

The convergence of the Rayleigh-Ritz Method in quantum chemistry

I. the criteria of convergence
  • Bruno Klahn
  • Werner A. Bingel
Original Investigations

Abstract

Two sufficient criteria for the convergence of the Rayleigh-Ritz Method (RRM) with respect to the eigenvalues (“E-convergence”) of non-relativistic electronic Hamiltonians of molecules are discussed and compared. Moreover, a necessary and sufficient criterion is given. By example (Sect. 9) it is shown that the L2-completeness of the basis is not sufficient to guarantee E-convergence. The convergence of the wave functions in different norms (“Ψ-convergence”) is also investigated. In particular, sufficient conditions for the one-particle basis functions (orbitals) are given, such that a CI calculation in this basis is both E- and Ψ-convergent.

Key words

Criteria of convergence Completeness of basis sets Energy space CI calculations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kato, T.: Trans. Am. Math. Soc. 70, 195 (1951)Google Scholar
  2. 2.
    Michlin, S. G.: Variationsmethoden der mathematischen Physik. Berlin: Akademie-Verlag 1962Google Scholar
  3. 3.
    Bonitz, G.: Zum Ritzschen Verfahren. Berlin: VEB Deutscher Verlag der Wissenschaften 1971Google Scholar
  4. 4.
    Klahn, B., Bingel, W. A.: Theoret. Chim. Acta, Part II, next paperGoogle Scholar
  5. 5.
    Weinstein, A., Stenger, W.: Methods of intermediate problems for eigenvalues. New York: Academic Press 1972Google Scholar
  6. 6.
    Kato, T.: Perturbation theory for linear operators. Berlin: Springer Verlag 1966Google Scholar
  7. 7.
    Simon, B.: Quantum mechanics for Hamiltonians defined as quadratic forms. Princeton: University Press 1971Google Scholar
  8. 8.
    Klahn, B.: Thesis, Universität Göttingen (1975)Google Scholar
  9. 9.
    Smirnow, W. I.: Lehrbuch der höheren Mathematik, Vol. V. Berlin: VEB Deutscher Verlag der Wissenschaften 1962Google Scholar
  10. 10.
    Eckart, C.: Phys. Rev. 36, 878 (1930)Google Scholar
  11. 11.
    Löwdin, P. O.: Advan. Chem. Phys. 2, 267 (1959)Google Scholar
  12. 12.
    Courant, R., Hilbert, D.: Methods of mathematical physics, Vol. I. New York: Interscience 1953Google Scholar
  13. 13.
    Klahn, B., Bingel, W. A.: to be published in Intern. J. Quantum Chem.Google Scholar
  14. 14.
    Luchka, A. Yu.: Ukr. Mat. Zh. 20, 713 (1968) (Engl. translation p. 617)Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • Bruno Klahn
    • 1
  • Werner A. Bingel
    • 1
  1. 1.Theoretical Chemistry GroupUniversity of GöttingenGöttingenFederal Republic of Germany

Personalised recommendations