Irreducible representations nondegenerate

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... 

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. 

One doesn t need a real degeneracy to benefit from this effect. Consider a nondegenerate two-level system, 84, with the two levels of different symmetry (here labeled A, B) in one geometry. If a vibration lowers the symmetry so that these two levels transform as the same irreducible representation, call it C, then they will interact, mix, and repel each other. For two electrons, the system will be stabilized. The technical name of this effect is a second order Jahn-Teller deformation.67... 

The/ orbitals will cluster into sets according to their irreducible representations. Thus (a)/ —> A] +T +T2 in Td symmetry, and there is one nondegenerate orbital and two sets of triply degenerate orbitals, (b) / —> A2u + Tiu + T2u, and the pattern of splitting (but not the order of energies) is the same. 

But as shown in Appendix XI, F must be a constant matrix, as also must F,i if and R "> > are identical irreducible representations. If, on the other hand, Rf" > and R > > are nonequivalont irreducible representations, then necessarily F,v = 0. Thus, in (8), Fn, F22, F23, and F33 would each bo constant matrices, while F12, F13 (also F21 and F31) would vanish. In this same example, if R were nondegenerate and R " triply degenerate d = 1, 4 = 3), F would have the detailed form... 

The stable (equilibrium) geometry corresponds to the nondegenerate electronic state, Sn(G ), described by the one-dimensional irreducible representation T(Sn). Otherwise the system continues in symmetry descent. 

The symmetry elements and reflection plane containing the principal axis, respectively. The superscript in the notations for the and irreducible representations (IRs) indicates the behavior of the IRs under the operation. Since the characters of the E operation in all IRs of are either 1 or 2, this means that all diatomic electronic states are spatially either nondegenerate or doubly degenerate. 

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