Atoms symmetry related degeneracies

The main effect is already taken into account if symmetry numbers are included in the densities of states. The symmetry number is a correction to the density of states that allows for the fact that indistinguishable atoms occupy symmetry-related positions and these atoms have to obey the constraints of the Pauli principle (i.e. the wave function must have a definite symmetry with respect to any permutation), whereas the classical density of states contains no such constraint. The density of states is reduced by a factor that is equal to the dimension of the rotational subgroup of the molecule. When a molecule is distorted, its symmetry is reduced, and so its symmetry number changes by a proportion that is equivalent to the number of indistinguishable ways in which the distortion may be produced. For example, the rotational subgroup of the methane molecule is T, whose dimension is 12, whereas the rotational subgroup of a distorted molecule in which one bond is stretched is C3, whose dimension is 3. The ratio of these symmetry numbers, 4, is the number of ways in which the distortion can occur, i.e. the reaction path degeneracy. 

While Hirsch conceived his 2(n + l)2 electron rule for spherical aromatics, subsets of three-dimensionally aromatic molecules having very high symmetries ( Ti, Oj, h, etc.), it can be applied to lower symmetry clusters such as the nine-vertex examples above. In cluster molecules the highest degeneracy MOs of a spherically harmonic atom set split into related, but lower degeneracy (or even non-degenerate) components. 

Up to now, the centres considered in this chapter were isolated atoms with cubic symmetry, but it has been seen in Chap. 6 that there exists many other donor centres with non-cubic symmetry. These centres, with symmetries lower than cubic, present an orientational degeneracy in addition to the electronic degeneracies related to their atomic structure. The effect of a uniaxial stress on their spectroscopic properties depends also on this additional degeneracy so that it cannot be treated as a whole. The general piezospectroscopic properties of non-cubic centres in cubic crystals have been discussed by Kaplyanskii [73]. 

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