Theoretica chimica acta

, Volume 42, Issue 2, pp 111–144 | Cite as

Quotients of group algebrae in the calculation of intermediate ligand field matrix elements

  • John C. Donini
  • Bryan R. Hollebone


The structure of the classes of symmetry elements excluded during the subduction of the representations of SU(2) onto the finite group 0* is shown to quantitatively define the relationship of the coupling algebrae of these two groups. This relationship is formalized as a quotient algebra. This quotient algebra is realized as 3Γ-like symbols which exist whether or not the quotient can be defined as a group. These symbols distribute the value of a reduced matrix element of SU(2) onto the subduced reduced matrix elements of O* and are termed Partition Coefficients. Since the structure of the excluded symmetry classes is independent of the quantization of O*, these Partition Coefficients can be used to define the values of the matrix elements of O* without reference to the form of its basis set. Thus, the choice of physical interpretation of the ligand field is unimportant. The strong field, weak field, Russell-Saunders and j-j coupling models are all unified in terms of the Partition Coefficients and the 3Γ symbols which are appropriate to the quantization.

Key words

Ligand field theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1. (a)
    Kibler, M.: Intern. J. Quantum Chem. 3, 795 (1969)Google Scholar
  2. 1. (b)
    Kibler, M.: J. Mol. Spectry. 26, 111 (1969)Google Scholar
  3. 2.
    König, E., Kremer, S.: Intern. J. Quantum Chem. 8, 347 (1974)Google Scholar
  4. 3.
    König, E., Kremer, S.: Theoret. Chim. Acta (Berl.) 32, 27 (1973)Google Scholar
  5. 4.
    Figgis, B. N.: Introduction to ligand fields. New York: Interscience 1966Google Scholar
  6. 5.
    Griffith, J. S.: Theory of transition metal ions. London: Cambridge University Press 1962Google Scholar
  7. 6.
    Griffith, J. S.: The irreducible tensor method for molecular symmetry groups. New Jersey: Prentice Hall, Englewood Cliffs 1962Google Scholar
  8. 7.
    Ballhausen, C. J.: Introduction to ligand field theory. New York: McGraw-Hill 1962Google Scholar
  9. 8.
    Hempel, J. C., Donini, J. C., Hollebone, B. R., Lever, A. B. P.: J. Am. Chem. Soc. 96, 169 (1974)Google Scholar
  10. 9.
    Donini, J. C., Hollebone, B. R., London, G., Lever, A. B. P., Hempel, J. C.: Inorg. Chem. 14, 455 (1975)Google Scholar
  11. 10.
    Hollebone, B. R., Donini, J. C.: Theoret. Chim. Acta (Berl.) 37, 233 (1975)Google Scholar
  12. 11.
    Donini, J. C., Hollebone, B. R.: Theoret. Chim. Acta (Berl.) 42, 97 (1976)Google Scholar
  13. 12.
    Matsen, F. A., Plummer, O. R. in: Group theory and its applications. Loebl, E. M. Ed., p. 221. New York: Academic Press 1968Google Scholar
  14. 13.
    Biedenharn, L. C.: J. Math. Phys. 2, 433 (1961)Google Scholar
  15. 14.
    Hollebone, B. R., Donini, J. C., Lever, A. B. P.: Mol. Phys. 22, 155 (1971)Google Scholar
  16. 15.
    Coleman, A. J. in: Group theory and its applications. Loebl, E. M. Ed. Vol. 1, p. 57. New York: Academic Press 1968Google Scholar
  17. 16.
    Lomont, J. S.: Application of finite groups. New York: Academic Press 1959Google Scholar
  18. 17.
    Wigner, E.: Quantum theory of angular momentum. Biedenharn, L. C., Van Dam, H. Eds. p. 89. New York: Academic Press 1965Google Scholar
  19. 18.
    Racah, G.: Phys. Rev. 76, 1352 (1949)Google Scholar
  20. 19.
    Donini, J. C., Hollebone, B. R., Lever, A. B. P.: Prog. Inorg. Chem. 22 (1975) in pressGoogle Scholar
  21. 20.
    Tang, Au-Chin, Sun, C. C., Kiang, Y. S., Deng, Z. H., Liu, J. C., Chiang, C. E., Yan, G. S., Goo, Z., Tai, S. S.: Scientia Sinica 15, 610 (1966)Google Scholar
  22. 21.
    Edmonds, A. R.: Proc. Roy. Soc. (London) A268, 567 (1962)Google Scholar
  23. 22.
    Fano, U., Racah, G.: Irreducible tensorial sets. New York: Academic Press 1959Google Scholar
  24. 23.
    Harnung, S. E., Schaffer, C. E.: Struct. Bonding 12, 201, 257 (1972)Google Scholar
  25. 24.
    Sugano, S., Tanabe, Y., Kamimura, H.: Multiplets of transition metal ions in crystals. New York: Academic Press 1970Google Scholar
  26. 25.
    Killingbeck, J.: J. Math. Phys. 11, 2268 (1970)Google Scholar
  27. 26.
    Nielson, C. W., Köster, G. F.: Spectroscopic coefficients for the p n,d n,f n configurations. Cambridge, Mass.: M.I.T. Press 1963Google Scholar
  28. 27.
    Judd, B. R.: Operator techniques in atomic spectroscopy. New York: McGraw-Hill 1963Google Scholar
  29. 28.
    Gerloch, M., Mackey, D. J.: J. Chem. Soc. (A) 2612 (1972)Google Scholar
  30. 29.
    Donini, J. C.: Thesis. York University, Toronto, Canada 1973Google Scholar
  31. 30.
    Rotenberg, M., Bivins, R., Metropolis, N., Wooten Jr., J. K.: The 3-j and 6-j symbols. Cambridge, Mass.: M.I.T. Press 1959Google Scholar
  32. 31.
    Hollebone, B. R.: Inorg. Chem. (in press)Google Scholar
  33. 32.
    Donini, J. C., Hollebone, B. R., Stillman, M. S.: 58th C.I.C. Conference, presentation No. 177. Toronto, Canada 1975Google Scholar
  34. 33.
    Donini, J. C.: In preparationGoogle Scholar
  35. 34.
    Ellzey, M. L.: Intern. J. Quantum Chem. 7, 253 (1973)Google Scholar
  36. 35.
    Soliverez, C. E.: Intern. J. Quantum Chem. 7, 1139 (1973)Google Scholar
  37. 36.
    Gerloch, M., Slade, R. C.: Ligand field parameters. London: Cambridge University Press 1973Google Scholar
  38. 37.
    Shibuya, T. I., Wulfman, C. E.: Proc. Roy. Soc. A 286, 376 (1965)Google Scholar
  39. 38.
    Hollebone, B. R., Donini, J. C.: J. Chem. Soc. Faraday II (1975) in pressGoogle Scholar
  40. 39.
    Wulfman, C. E.: Group theory and its applications. Loebl, E. M. Ed. Vol. 2, p. 145. New York: Academic Press 1968Google Scholar
  41. 40.
    For O and Oh this can be proven by construction: J. C. Donini, unpublished resultsGoogle Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

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

Personalised recommendations