Theoretica chimica acta

, Volume 37, Issue 3, pp 233–250 | Cite as

The coupling algebra of trigonally subduced crystal field eigenvectors

  • Bryan R. Hollebone
  • J. C. Donini
Commentationes

Abstract

The general theory of subduction of eigenvectors between infinite groups is used to derive a finite group subduction operator and define the corresponding subduction coefficients. The coupling behaviour of these subduced eigenvectors can then be described in terms of 3 Γ symbols. These symbols, defined only in relation to complex basis sets are all fully real and have all phases fixed by the subduction operator. They differ from V coefficients in two phase relationships and have the advantage, unlike V coefficients, of retaining all the symmetry properties and selection rules of Wigner 3-j symbols. Appropriate label systems which render these properties in terms of simple algebras are given for all quantizing axes available in O h . The specific set of 3Γ symbols for each quantization is determined by the orientation of the coordinate axes in the Hamiltonian. The four possible orientations for trigonal quantization are examined and the operator chosen which produces eigenvectors with conventional conjugate phases and a fully real set of 3Γ symbols.

Key word

Crystal field theory 

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References

  1. 1.
    Lever,A.B.P., Hollebone,B.R.: J. Am. Chem. Soc. 94, 1816 (1972)Google Scholar
  2. 2.
    Hollebone,B.R., Lever,A.B.P., Donini,J.C.: Mol. Phys. 22, 155 (1971)Google Scholar
  3. 3.
    Hemple,J.C., Donini,J.C., Hollebone,B.R., Lever,A.B.P.: J. Am. Chem. Soc. 96, 169 (1974)Google Scholar
  4. 4.
    Judd,B.R.: Operator techniques in atomic spectroscopy. New York: McGraw-Hill 1963Google Scholar
  5. 5.
    Matsen,F.A., Plummer,O.R.: In: Group theory and its applications, p. 221. E.M. Loebl, Ed. New York: Academic Press 1968Google Scholar
  6. 6.
    Donini,J.C., Hollebone,B.R., Lever,A.B.P.: J. Am. Chem. Soc., 93, 6455 (1971)Google Scholar
  7. 7.
    Condon,E.U.,Shortley,G.H.: The theory of atomic spectra. Cambridge 1970Google Scholar
  8. 8.
    Sugano,S., Tanabe,Y., Kamimura,H.: Multiplets of transition metal ions in crystals. New York: Academic Press 1970Google Scholar
  9. 9.
    Cotton,F.A.: Chemical applications of group theory. New York: Interscience 1963Google Scholar
  10. 10.
    Koster,G.F.: Notes on group theory: Solid State and Molecular Theory Group, Technical Report 8. Cambridge, Massachusetts: Massachusetts Institute of Technology 1956Google Scholar
  11. 11.
    Ellzey,M.L.: Intern. J. Quantum Chem. 7, 253 (1973)Google Scholar
  12. 12. a.
    Ballhausen,C.J.: Introduction to ligand field theory. New York: McGraw-Hill 1962Google Scholar
  13. 12. b.
    Griffith,J.S.: The theory of transition metal ions. London: Cambridge University Press 1962Google Scholar
  14. 13.
    Griffith,J. S.: The Irreducible Tensor Method for Molecular Symmetry Groups, New Jersey: Prentice-Hall, Englewood Cliffs 1962Google Scholar
  15. 14.
    Harnung,S. E., Schäffer,C.E.: Struct. Bonding 12, 201 (1972). Note that these authors define 3Γ symbols on the real basis while that discussed here is on the complex basis. On the real basis the combinatorial algebra for components is lost.Google Scholar
  16. 15.
    Birss,F. R.: private communicationGoogle Scholar

Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • Bryan R. Hollebone
    • 1
  • J. C. Donini
    • 1
  1. 1.Department of ChemistryUniversity of AlbertaEdmontonCanada

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