MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.5.62 Lecture #13 Supplement:Nuclear Spin and Symmetry Number P12Ψ = ±Ψ is required for all pairs of identical particles. + for boson, – for fermion.(Identical means same nucleus, not same chemical environment.) (P12)2Ψ = Ψ means that there are only two classes of symmetry functions for eachgroup of identical nuclei. e.g. benzene 6H Ii = 1/2 I = 3,1 para: degeneracy 7 + 3 = 10 4, 2, 0 ortho: degeneracy 9 + 5 + 1 = 15 6C Ii = 0 only Itot = 0 To construct Ψ with legal permutation symmetry with respect to ψrot, we need to considersuperpositions of bonded networks. e.g. the C2 symmetry operation permutes pairs of C’sand pairs of H’s. Yet the permutation must be even with respect to C’s. How many bonded networks are there? 6!6! 2·6 We divide by 12 because this is the overcounting due to rotationally related identicalstructures. 12 is the symmetry number. We divide qrot by the symmetry number because we are correcting the number of distinctbonded networks. We normally do not include the number of bonded networks becausethis is merely a function of the total number of identical atoms. This permutationdegeneracy is conserved as long as atoms are conserved. But actually every rotation-vibration level of benzene is 6!6! 12 = 43,200 fold degenerate!But we never worry about this degeneracy because, for normal levels of internalexcitation, bonds are conserved. But at high excitation, bond-breaking isomerization occurs. This means that the appropriate symmetry group for a molecule is called “The MolecularSymmetry Group”, a subgroup of the “Complete Permutation Inversion Group”. The subgroup consists of all “feasible” permutations and
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