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Theoretica chimica acta

, Volume 65, Issue 5–6, pp 317–363 | Cite as

Phase-fixed double-group 3-Γ symbols. I. A novel exposition of the general theory of 3-Γ symbols and coupling coefficients

  • Ture Damhus
  • Sven E. Harnung
  • Claus E. Schäffer
Original Investigations

Abstract

The present paper is the first in a series aiming at the establishment of a transparent and readily applicable Wigner-Racah algebra for all the noncommutative double groups.

Starting from the Wigner-Eckart theorem in a very general setting, the theory of the fundamental quantities called here triple coefficients — and the closely related coupling coefficients — is developed and leads through a careful discussion of permutational symmetries to the concept of 3symbols. By basing the exposition on triple coefficients and by consistently using matrix representations, we obtain a notation and a terminology which enable a clear separation of permutational properties and problems concerning complex conjugation, and a more transparent discussion of tensor (Kronecker) product multiplicities.

A particularly elegant formalism is obtained for a situation which generalizes that of the classical rotation-group Wigner-Racah algebra, viz., in which there is a fixed group element effecting (through the inner automorphism it defines) complex conjugation of all the standard irreducible matrix representations.

Key words

three-gamma symbols coupling coefficients triple coefficients complex conjugation of irreducible matrix representations inner automorphisms Frobenius-Schur classification Derome-Sharp matrices Wigner-Racah algebra 

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References

  1. 1.
    Griffith, J. S.: The Irreducible Tensor Method for Molecular Symmetry Groups, Englewood Cliffs, N.J.: Prentice-Hall, 1962Google Scholar
  2. 2.
    Harnung, S. E., Schäffer, C. E.: Struct. Bond. 12, 257 (1972)Google Scholar
  3. 3.
    Schäffer, C. E.: Physica 114A, 28 (1982)Google Scholar
  4. 4.
    Brorson, M., Damhus, T., Schäffer, C. E.: Comments Inorg. Chem., 3, 1 (1984).Google Scholar
  5. 5. (a)
    Andersen, P., Damhus, T., Pedersen, E., Petersen, A.: to appear in Acta Chem. Scand.Google Scholar
  6. 5. (b)
    Damhus, T., Pedersen, E.: Inorg. Chem. 23, 695 (1984);Google Scholar
  7. 5. (c)
    and references in papers (a) and (b)Google Scholar
  8. 6.
    Damhus, T.: Molec. Phys. 50, 497 (1983)Google Scholar
  9. 7.
    Wigner, E. P.: Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, New York: Academic Press, 1959Google Scholar
  10. 8.
    Fano, U., Racah, G.: Irreducible Tensorial Sets, New York: Academic Press, 1959Google Scholar
  11. 9.
    Rotenberg, M., Bivins, R., Metropolis, N., Woolen, Jr., J. K.: The 3-j and 6-j symbols, MIT Press, 1959Google Scholar
  12. 10.
    Biedenharn, L. C., van Dam, H., Eds.: Quantum Theory of Angular Momentum. A Collection of Reprints and Original Papers, New York: Academic Press, 1965Google Scholar
  13. 11.
    Silver, B. L.: Irreducible Tensor Methods. An Introduction for Chemists, New York: Academic Press, 1976Google Scholar
  14. 12.
    Biedenharn, L. C., Louck, J. D.: The Racah-Wigner algebra in Quantum Theory, Vol. 9 of Encyclopedia of Mathematics and its Applications. Massachusetts: Addison-Wesley, 1981Google Scholar
  15. 13.
    Normand, H.-M.: A Lie Group: Rotations in Quantum Mechanics, Berlin: Springer 1980Google Scholar
  16. 14.
    Wigner, E. P.: Unpublished manuscript, 1940; reprinted in [10]Google Scholar
  17. 15.
    Derome, J.-R., Sharp, W. T.: J. Math. Phys. 6, 1584 (1965)Google Scholar
  18. 16.
    Derome, J.-R.: J. Math. Phys. 7, 612 (1966)Google Scholar
  19. 17.
    Butler, P. H.: Philos. Trans. Roy. Soc. London A277, 545 (1975)Google Scholar
  20. 18.
    Kibler, M. R.: J. Mol. Spectrosc. 62, 247 (1976)Google Scholar
  21. 19.
    Butler, P. H.: in Symmetries in Science, B. Gruber, R. S. Millman, Eds. pp. 89–104. New York: Plenum 1980Google Scholar
  22. 20.
    Harnung, S. E., Schäffer, C. E.: Struct. Bond. 12, 201 (1972)Google Scholar
  23. 21.
    Damhus, T.: Licentiate (Ph.D.) thesis, University of Copenhagen 1984Google Scholar
  24. 22.
    Butler, P. H.: Point Group Symmetry Applications. New York: Plenum 1981Google Scholar
  25. 23.
    Edmonds, A. R.: Angular Momentum in Quantum Mechanics, Vol. 4 of Investigations in Physics, E. P. Wigner, R. Hofstadt, Eds. N.J.: Princeton University Press 1957Google Scholar
  26. 24.
    Meijer, P. H. E., Bauer, E.: Group Theory. The Application to Quantum Mechanics. Amsterdam: North-Holland 1962Google Scholar
  27. 25.
    McWeeny, R.: Symmetry: An Introduction to Group Theory and its Applications, The International Encyclopedia of Physical Chemistry and Chemical Physics, Mathematical Techniques, Vol. 3. Pergamon/MacMillan 1963Google Scholar
  28. 26.
    Tinkham, M.: Group Theory and Quantum Mechanics, New York: McGraw-Hill 1964Google Scholar
  29. 27.
    Brink, D. M., Satchler, G. R.: Angular Momentum, 2nd Ed., Oxford: Oxford University Press 1971Google Scholar
  30. 28.
    Miller, Jr., W.: Symmetry Groups and Their Applications. New York: Academic Press 1972Google Scholar
  31. 29.
    Janssen, T.: Crystallographic Groups. Amsterdam: North-Holland 1973Google Scholar
  32. 30.
    Chisholm, C. D. H.: Group Theoretical Techniques in Quantum Chemistry. New York: Academic Press 1976Google Scholar
  33. 31.
    Schensted, I. V.: A Course on the Application of Group Theory to Quantum Mechanics. Peaks Island, Maine: NEO Press 1976Google Scholar
  34. 32.
    Judd, B. R.: Amer. J. Phys. 49, 371 (1981)Google Scholar
  35. 33.
    Petraschen, M. J., Trifonov, E. D.: Applications of Group Theory in Quantum Mechanics. MIT Press 1969Google Scholar
  36. 34.
    Wigner, E. P.: SIAM J. Appl. Math. 25, 169 (1973)Google Scholar
  37. 35.
    Klein, D. J.: in Group Theory and its Applications III, pp. 1–93, Loebl, E. M., Ed. New York: Academic Press 1975Google Scholar
  38. 36.
    van den Broek, P. M., Cornwell, J. F.: Phys. Stat. Sol. (b) 90, 211 (1978)Google Scholar
  39. 37.
    Agrawala, V. K., Belinfante, J. G.: Ann. Phys. (New York) 49, 130 (1968)Google Scholar
  40. 38. (a)
    Klimyk, A. U.: Teor. Mat. Fiz. (Russ.) 8, 55 (1971) = Theor. Math. Phys. (transl.) 8, 668 (1971);Google Scholar
  41. 38. (b)
    Klimyk, A. U.: Teor. Mat. Fiz. (Russ.) 13, 327 (1972) = Theor. Math. Phys. (transl.) 13, 1171 (1972);Google Scholar
  42. 38. (c)
    Klimyk, A. U.: Rep. Math. Phys. 7, 153 (1972)Google Scholar
  43. 39.
    Mezincescu, L.: J. Math. Phys. 18, 453 (1977)Google Scholar
  44. 40.
    Agrawala, V. K.: J. Math. Phys. 21, 1562 (1980)Google Scholar
  45. 41.
    Schäffer, C. E.: Proc. Roy. Soc. London A297, 96 (1967)Google Scholar
  46. 42. (a)
    Atkins, P. N., Child, M. S., Philips, C. S. G.: Tables for Group Theory. Oxford: Oxford University Press 1970;Google Scholar
  47. 42. (b)
    Salthouse, J. A., Ware, M. J.: Point group character tables and related data. Cambridge: University Press 1972Google Scholar
  48. 43.
    Biedenharn, L. C., Brouwer, W., Sharp, W. T.: The algebra of representations of some finite groups. Rice University Studies 54, no. 2, 1968Google Scholar
  49. 44.
    van Zanten, A. J., de Vries, E.: J. Algebra 25, 475 (1973)Google Scholar
  50. 45.
    Butler, P. H., King, R. C.: Canad. J. Math. 26, 328 (1974)Google Scholar
  51. 46.
    Derome, J.-R.: J. Math. Phys. 8, 714 (1967)Google Scholar
  52. 47.
    Dirl, R.: J. Math. Phys. 20, 659 (1979); Dirl, R.: ibid. 20, 1562 (1979)Google Scholar
  53. 48.
    Derome, J.-R., Jakimov, G.: Canad. J. Phys. 48, 2169 (1970)Google Scholar
  54. 49.
    van Zanten, A. J., de Vries, E.: Canad. J. Math. 26, 1090 (1974); ibid. 27, 528 (1975)Google Scholar
  55. 50.
    Harnung, S. E.: Molec. Phys. 26, 473 (1973)Google Scholar
  56. 51.
    König, E., Kremer, S.: Z. Naturforsch. 29a, 1179 (1974)Google Scholar
  57. 52.
    Kibler, M. R., Guichon, P. A. M.: Intern. J. Quant. Chem. 10, 87 (1976)Google Scholar
  58. 53.
    Newmarch, J. D.: J. Math. Phys. 24, 757 (1983)Google Scholar
  59. 54.
    Damhus, T.: J. Math. Phys. 22, 7 (1981)Google Scholar
  60. 55.
    Frobenius, F. G., Schur, I: Sitzungsber. Königl. Preuss. Akad. Wiss. Berlin, 186 (1906)Google Scholar
  61. 56.
    Damhus, T.: Linear Algebra Appl. 32, 125 (1980)Google Scholar
  62. 57.
    Butler, P. H.: in Recent Advances in Group Theory and Their Application to Spectroscopy, pp. 123–177. Donini, J. C., Ed. New York: Plenum 1979Google Scholar
  63. 58.
    Damhus, T.: Double groups as symmetry groups for spin-orbit coupling Hamiltonians; submitted to MatchGoogle Scholar
  64. 59.
    Damhus, T., Harnung, S. E.; in preparationGoogle Scholar
  65. 60.
    Coleman, A.: Adv. Quant. Chem. 4, 83 (1968)Google Scholar
  66. 61.
    Kerber, A.: Representations of permutation groups I+II (I: Lecture Notes in Mathematics 240 (1971), II: ibid. 495 (1975), Berlin: Springer-Verlag)Google Scholar
  67. 62.
    Chatterjee, R., Lulek, T.: J. Math. Phys. 23, 922 (1982)Google Scholar
  68. 63.
    Winck, D. E.: J. Math. Phys. 17, 1166 (1976)Google Scholar
  69. 64.
    Isaacs, I. M.: Character Theory of Finite Groups. New York: Academic Press 1976Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Ture Damhus
    • 1
  • Sven E. Harnung
    • 1
  • Claus E. Schäffer
    • 1
  1. 1.Chemistry Department I, H. C. Ørsted InstituteUniversity of CopenhagenCopenhagen ØDenmark

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