Theoretica chimica acta

, Volume 73, Issue 4, pp 247–277 | Cite as

Permutation representations in molecular symmetry

  • Gerhard Fieck


The reducible representations of the point groups are generally studied because of their relevance to molecular orbital and vibration theory. Triple correlations within the polyhedra are described by group-theoretical invariants that are related to the permutation representations and termed polyhedral isoscalar factors. These invariants are applied in theorems on matrix elements referring to the symmetry-adapted bases at different centres. Further invariants or geometrical weight factors inter-relate different types of reduced matrix elements of irreducible tensors (generalization of the Wigner-Eckart theorem to the polycentric case). As a demonstration a complete tabulation is given for the point group C.


Polycentric molecules Permutational representations Symmetry adaption Irreducible tensors Wigner-Eckart theorem 


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References and notes

  1. 1.
    Louck JD, Galbraith HW (1976) Eckart vectors, Eckart frames, and polyatomic molecules. Rev Mod Phys 48:69Google Scholar
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    McWeeny R (1963) Symmetry, chap 65. Pergamon Press, OxfordGoogle Scholar
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    Fieck G (1978) Symmetry adaption IV. The force constant matrix of symmetric molecules. Theor Chim Acta 49:211Google Scholar
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    Fieck G (1977) Symmetry adaption reduced to tabulated quantities. Theor Chim Acta 44:279Google Scholar
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    Fieck G (1982) Symmetry of polycentric systems. Springer, Berlin Heidelberg New YorkGoogle Scholar
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    For instance in: Wybourne BG (1974) Classical groups for physicists, chap 19.8. Wiley, New YorkGoogle Scholar
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    Racah G (1949) Theory of complex spectra IV. Phys Rev 76:1352; cf. also the preceeding reference, chap 19.14Google Scholar
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    Fieck G (1978) Symmetry adaption II. Representations on symmetric polyhedra. Theor Chim Acta 49:187Google Scholar
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    Griffith JS (1962) The irreducible tensor method for molecular symmetry groups. Prentice-Hall, Englewood Cliffs (NJ)Google Scholar
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    Piepho SB, Schatz PN (1983) Group theory in spectroscopy, chap 19. Wiley, New YorkGoogle Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Gerhard Fieck
    • 1
  1. 1.Schule Marienau Nr. 11DahlemFederal Republic of Germany

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