Representations of the Symmetric Group Generated by the Spin Eigenfunctions
In the previous chapter we have seen that the different irreducible representations of the symmetric group can be characterized by the different Young shapes. The notion of the Young tableau was helpful for the construction of the orthogonal and natural representation. In this chapter we shall consider the behavior of the spin eigenfunctions under the operations of the permutations of the electronic coordinates, and we shall show that they generate irreducible representations of the symmetric group; the latter can be characterized by Young shapes having not more than two rows. We shall also establish a one-to-one correspondence between the Young tableaux and the functions generated in the different methods. The representation matrices will play an important role in the calculation of the matrix elements of the Hamiltonian; the one-to-one correspondence gives us new and effective methods for the construction of spin eigenfunctions.
KeywordsRepresentation Matrix Transformation Matrix Symmetric Group Representation Matrice Young Tableau
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- 1.T. Yamanouchi, Proc. Phys. Math. Soc. Jpn. 18, 623 (1936).Google Scholar
- 2.M. Kotani, A. Amemiya, E. Ishiguro, and T. Kimura, Tables of Molecular Integrals, 2nd ed., Maruzen, Tokyo (1963), Chapter 1.Google Scholar
- 3.R. Pauncz, Alternant Molecular Orbital Method, W. B. Saunders, Philadelphia (1967), Appendix 2, p. 216.Google Scholar
- 6. (a).T. Yamanouchi, Proc. Phys. Math. Soc. Jpn. 19, 436 (1937)Google Scholar
- 6. (b).
- 10.E. M. Corson, Perturbation Methods in the Quantum Mechanics of n-Electron Systems, Blackie and Son, London (1951), p. 217.Google Scholar
- 11.L. F. Mattheiss, Quart. Progr. Rep., Solid State Molecular Theory Group MIT 34 58, (1959).Google Scholar
- 12. (a).
- 12. (b).
- 15.D. E. Littlewood, The Theory of Group Characters, Clarendon Press, Oxford (1950), p. 94.Google Scholar
- 16.I. G. Kaplan, Sov. Phys.-JETP 14, 401 (1962).Google Scholar