Nondegenerate representations

In symmetry point groups in which x, y and 2 belong to separate nondegenerate representations, a naive extension of the analogy with orbital motion might suggest that the three spin functions associated with the triplet yl i, Aq ) should combine similarly to functions that transform like x, y and 2 , but this is not the case. Due to the unique nature of spin, the three spin functions Ax Ay and A have to be assigned to the irreps of Rx Ry and R respectively [2, p. 206]. 

It is more difficult to show that the two remaining orbitals are still degenerate also, but they cannot belong to the non-degenerate representations. The nondegenerate representations are either symmetric or antisymmetric under 90° rotation about any of the Q axes, and neither the d i nor the d y satisfy this criterion. There is only one doubly degenerate representation with g symmetry the Cg, and this is the representation for these last two d orbitals. 

Theorem 1 The 2-RDM for the antisymmetric, nondegenerate ground state of an unspecified N-particle Hamiltonian H with two-particle interactions has a unique preimage in the set of N-ensemble representable density matrices D. 

The first system to be discussed is ethene. Ethene is a closed-shell molecule with a point group that only includes nondegenerate irreducible representations. Its MCD spectrum therefore can only include terms. Despite this restriction, the ethene MCD spectrum is a useful testing ground and many of the insights obtained can be usefully apphed to other MCD calculations. 

Therefore, two of the four lowest it energy levels are doubly degenerate and the other two are nondegenerate moreover, since no irreducible representation occurs more than once in (9.87), we can obtain the correct linear combinations for the six lowest it MOs without solving a secular equation. 

We can now show that the eigenfunctions for a molecule are bases for irreducible representations of the symmetry group to which the molecule belongs. Let us take first the simple case of nondegenerate eigenvalues. If we take the wave equation for the molecule and carry out a symmetry operation, / , upon each side, then, from 5.1-1 and 5.1-2 we have... 

This can easily be demonstrated, using direct product representations for orbitals alone, when only nondegenerate orbitals are involved. The same is true when degenerate orbitals are involved, but more sophisticated methods of proof are required. 

We list here full matrix representations for several groups. Abelian groups are omitted, as their irreps are one-dimensional and hence all the necessary information is contained in the character table. We give C3v (isomorphic with D3) and C4u (isomorphic with D4 and D2d). By employing higher 1 value spherical harmonics as basis functions it is straightforward to extend these to Cnv for any n, even or odd. We note that the even n Cnv case has four nondegenerate irreps while the odd n Cnv case has only two. 

Payne, S. E. Symmetric representations of nondegenerate generalized n-gons, Proc. Amer. Math. Soc. 19, 1321-1326 (1968)... 

A more general yet tractable approach to semi-Markov models is the phase-type distribution developed by Neuts [363], who showed that any nondegenerate distribution / (a) of a retention time A with nonnegative support can be approximated, arbitrarily closely, by a distribution of phase type. Consequently, all semi-Markov models in the recent literature are special phase-type distribution models. However, the phase-type representation is not unique, and in any case it will be convenient to consider some restricted class of phase-type distributions. 

From this it follows that the character under E is always the dimension of the given irreducible representation. The one-dimensional representations are nondegenerate and the two- or higher-dimensional representations are degenerate. The meaning of degeneracy will be discussed in Chapter 6. 

As environmental symmetry decreases, the orbitals will become split to an increasing extent. In the C2v point group, for example, all atomic orbitals will be split into nondegenerate levels. This is not surprising since the C2v character table contains only one-dimensional irreducible representations. This result shows at once that there are no degenerate energy levels in this point group. This has been stressed in Chapter 4 in the discussion of irreducible representations. 

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