Theoretica chimica acta

, Volume 42, Issue 2, pp 97–110 | Cite as

The coupling symbols and algebrae of subducible, octahedrally projected ligand field eigenvectors

  • John C. Donini
  • Bryan R. Hollebone


The 3Γ symbols required for the application of the Wigner-Eckart theorem to strong ligand field matrix elements are derived for complex basis functions quantized on the C 4 Z , C 3 XYZ , C 2 Z and C 2 XY axes of an octahedron. This scheme provides a standardized analysis technique for the matrix elements of subgroups in each of the four physically significant chains of the double group O h * . This standardization yields the minimum necessary number of ligand field parameters in any subgroup and makes possible the direct comparability of equivalent parameters in different symmetries. A unique numerical labelling for both representations and complex components on each axis provides both a simple component selection rule algebra and numerical phase factors governing permutation and conjugation of the 3Γ symbols.

Key words

Ligand field eigenvectors 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Hollebone, B. R., Donini, J. C.: Theoret. Chim. Acta. (Beil.) In PressGoogle Scholar
  2. 2.
    Hempel, J. C., Donini, J. C., Hollebone, B. R., Lever, A. B. P.: J. Am. Chem. Soc. 96, 1693 (1974)Google Scholar
  3. 3.
    Ballhausen, C. J.: Introduction to ligand field theory, p. 61. New York: McGraw-Hill 1962Google Scholar
  4. 4.
    Harnung, S. E., Schaffer, C. E.: Struct. Bonding 12, 201 (1972)Google Scholar
  5. 5.
    Griffith, J. S.: The irreducible tensor method for molecular symmetry groups. Englewood Cliffs, N.J.: Prentice-Hall 1962Google Scholar
  6. 6.
    Sugano, S., Tanabe, Y., Kamimura, H.: Multiplets of transition metal ions in crystals. New York: Academic Press 1970Google Scholar
  7. 7.
    Hollebone, B. R., Lever, A. B. P., Donini, J. C.: Mol. Phys. 22, 155 (1971)Google Scholar
  8. 8.
    Wigner, E. P.: Quantum theory of angular momentum, Biedenharn, L. C., Van Dam, H., Eds., pp. 80, 89. New York: Academic Press 1965Google Scholar
  9. 9.
    Matsen, F. A., Pluramer, O. R.: Group theory and its applications, Loebe, E. M., Ed., Vol. 1, p. 221. New York: Academic Press 1968Google Scholar
  10. 10.
    Dobosh, P.: Phys. Rev. A5, 2376 (1972)Google Scholar
  11. 11.
    Donini, J. C., Hollebone, B. R.: Unplublished. Tables of subduction coefficients are available from authorsGoogle Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • John C. Donini
    • 1
  • Bryan R. Hollebone
    • 2
  1. 1.Department of ChemistrySt. Francis Xavier UniversityAntigonishCanada
  2. 2.Department of ChemistryUniversity of AlbertaEdmontonCanada

Personalised recommendations