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A Handbook of Magnetochemical Formulae by Boca, Roman (Auth.) PDF

By Boca, Roman (Auth.)

ISBN-10: 012416014X

ISBN-13: 9780124160149

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7 Symmetry Group The Pauli principle demands that the complete state function be antisymmetric with respect to interchange of any two electrons. In the case of a two-electron problem, orbital and spin functions can be written separately. pffiffiffi For S51 the spin function jm1m2i is symmetric, being jα1α2i, ðjα1 β 2 i 1 jβ 1 α2 iÞ= 2 and jβ 1β 2i; the orbital part is then pffiffiffi antisymmetric. For S 5 0 the spin function is antisymmetric, ðjα1 β 2 i 2 jβ 1 α2 iÞ= 2; but the orbital part must be symmetric.

The latter demand follows from the fact that the representations of S half-integral in the group SU2S11 are double-valued. In all groups, including SN, the direct product of two identical IRs contains the fully symmetric representation Γi 3 Γj 5 Γred 5 Γ1 1 ? ð1:156Þ In symmetry groups SN, the direct product of any IR with its conjugate IR contains the totally antisymmetric representation Γi 3 Γ~ j 5 Γred 5 Γ2 1 ? This result can be proven by writing the decomposition formula X ni U χðRc ÞjΓi χðRc ÞjΓ 5 Γa 3 Γb 5 ð1:157Þ ð1:158Þ i with the integer coefficients 1X hc U χà ðRc ÞjΓi U χðRc ÞjΓ 5 Γa 3 Γb ni 5 g c ð1:159Þ Molecular Symmetry 49 where the summation runs over all classes (c) of the group; hc À order of the class; g À order of the group; the character of the reducible representation for the direct product is simply χðRc ÞjΓ 5 Γa 3 Γb 5 χðRc ÞjΓa UχðRc ÞjΓb ð1:160Þ For instance, in the S4 group there are decompositions ½4Š 3 ½14 Š 5 ½14 Š ð1:161Þ ½31Š 3 ½212 Š 5 ½14 Š 1 ½22 Š 1 ½31Š 1 ½212 Š ð1:162Þ ½22 Š 3 ½22 Š 5 ½14 Š 1 ½4Š 1 ½22Š ð1:163Þ and all of them contain the totally antisymmetric representation [14] 5 Γ2.

22 Young Tableaux Partitiona Shape Standard Tableaux with Corresponding Yamanouchi Symbolsb dim(Γi) 12 1, Γ1 [λ1λ2. e. zeros could be dropped, and repeated number given in exponent. b Each row of the Young diagram corresponds to a definite length of a cycle; each cycle has its row. a Molecular Symmetry 41 Within the group SN the following items hold true: 1. The partition [λ1λ2. λr] corresponds to a unique IR of the group. 2. The dimension of a specific IR is equal to the number of standard tableaux that can be constructed from the corresponding partition.

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A Handbook of Magnetochemical Formulae by Boca, Roman (Auth.)

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