Abstract
For all the subgroup hierarchies descending from the octahedral double group O *, we have obtained sets of 3-Γ symbols and discuss here their properties. We have entirely real sets of 3-Γ symbols for the tetrahedral and tetragonal hierarchies as well as for O * ⊃ C *3 . For the latter hierarchy and the tetragonal ones, formalisms almost as powerful as the classical one for the rotation group may be established. We also discuss results obtained for cases with strict adaption to D *3 where it is now known that non-real 3-Γ symbols are unavoidable.
The 3-Γ symbols are phase-fixed by the specification of basis functions (or, equivalently, subduction coefficients) generating them.
The significance of the concept of associated representations of O* is discussed. The problems raised by the two multiplicity triples UT1U and UT2U in O* are given particular attention.
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Damhus, T., Harnung, S. E., Schäffer, C. E.: paper I in this series
Same, paper II in the series
Harnung, S. E., Schäffer, C. E.: Struct. Bond. 12, 201 (1972)
Harnung, S. E.: Molec. Phys. 26, 473 (1973)
Joensen, F., Schäffer, C. E.: in preparation
Grenet G., Guichon, P., Kibler, M.: Recent advances in group theory and their application to spectroscopy, pp. 1–96, Donini, J. C. ed., New York: Plenum Press 1979
Damhus, T.: Linear Algebra Appl. 32, 125 (1980)
Frobenius, F. G.: Sitzungsber. Königl. Preuss. Akad. Wiss. (Berlin), 681 (1911)
Clifford, A. H.: Ann. Math. 38, 533 (1937)
Griffith, J. S.: The irreducible tensor method for molecular symmetry groups. Englewood Cliff, N.J.: Prentice-Hall 1962
Lulek, B., Lulek, T.: Rep. Math. Phys. 8, 321 (1975)
Lulek, B., Lulek, T.: Rep. Math. Phys. 8, 335 (1975)
Butler, P. H., Ford, A. M.: J. Phys. A 12, 1357 (1979)
Butler, P. H.: Point group symmetry applications. New York: Plenum Press 1981
Damhus, T., Harnung, S. E., Schäffer, C. E.: paper III in this series
Damhus, T., Harnung, S. E.: in preparation
Wigner, E. P.: Group theory and its application to the quantum mechanics of atomic spectra. New York: Academic Press 1959
Rotenberg, M., Bivins, R., Metropolis, N., Wooten, Jr., J. K.: The 3-j and 6-j symbols. MIT Press 1959
Fano, U., Racah, G.: Irreducible tensorial sets. New York: Academic Press 1959
Damhus, T.: licentiate (Ph.D.) thesis. University of Copenhagen, 1984
Griffith, J. S.: The theory of transition metal ions. Cambridge: University Press 1971; first impression 1961
Piepho, S. B.: pp. 405–491 in the volume edited by J. C. Donini (see [6] above)
Ceulemans, A., Beyens, D.: Phys. Rev. A 27, 621 (1983)
Damhus, T., Harnung, S. E., Schäffer, C. E.: paper IV in this series
Same, paper VI in the series
Damhus, T.: J. Math. Phys. 22, 7 (1981)
Boyle, L. L., Schäffer, C. E.: Intern. J. Quantum Chem. 8, 153 (1974)
Bickerstaff, R. P., Damhus, T.: submitted to Intern. J. Quantum Chem.
Damhus, T.: Double groups as symmetry groups for spin-orbit coupling Hamiltonians, submitted to Match
Lulek, B. Lulek, T., Szczepaniok, B.: Acta Phys. Polon. A 54, 545 (1978)
Lulek, B., Lulek, T.: Acta Phys. Polon. A 54, 561 (1978)
Bkllaszak, M.: Phys. Stat. Sol. B 109, 195 (1982)
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Damhus, T., Harnung, S.E. & Schäffer, C.E. Phase-fixed double-group 3-Γ symbols. V. 3-Γ symbols and coupling coefficients for all the octahedral double group-subgroup hierarchies. Theoret. Chim. Acta 65, 419–432 (1984). https://doi.org/10.1007/BF00550838
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DOI: https://doi.org/10.1007/BF00550838