Representation, degenerate reducible

The 2px and 2py orbitals of the two atoms together form a representation that reduces to irg and it,. These correspond to two doubly degenerate it orbitals, one of which lies in the yz plane... 

Distribution of the two additional electrons to 8 required for dianion formation among the degenerate LUMO orbitals of E, symmetry gives rise to four new states, since, within the 5v symmetry group, the direct product Ej Ej may be reduced to a sum of Aj, and 2 irreducible representations. The A2 state represents a triplet, while Aj and 2 are singlet states. 

Some of Fock s terminology may be mysterious to the modern reader. In particular, degenerate energy levels are energy eigenvalues whose eigenspaces are reducible (i.e., not irreducible) representations. 

We conclude that when the representation generated by the degenerate wave functions is reducible, the degenerate wave functions can always be chosen so that they divide up into subsets whose members transform... 

But this is just the expression that gives the elements of a matrix which is the product./ of two other matrices. Thus the matrices that describe the transformations of a set of k eigenfunctions corresponding to a /c-fold degenerate eigenvalue are a A-dimensional representation for the group. Moreover, this representation is irreducible. If it were reducible we could divide the k eigenfunctions. . . , or k linear combinations thereof, up... 

The symmetry of the normal mode of vibration that can take the molecule out of the degenerate electronic state will have to be such as to satisfy Eq. (6-7). The direct product of E with itself (see Table 6-11) reduces to A + A 2 + E. The molecule has three normal modes of vibration [(3 x 3) - 6 = 3], and their symmetry species are A + E. A totally symmetric normal mode, A, does not reduce the molecular symmetry (this is the symmetric stretching mode), and thus the only possibility is a vibration of E symmetry. This matches one of the irreducible representations of the direct product E E therefore, this normal mode of vibration is capable of reducing th eZ)3/, symmetry of the H3 molecule. These types of vibrations are called Jahn-Teller active vibrations. 

The GT Calculator has been designed to deal with real reducible representations, in which both the components of any given separably degenerate irreducible representations, such as E in C2h appear with equal weight. Data input and output conventions ensure that only such real representations are used. Thus, all the relevant worksheet input cells for these calculations are linked and the second member of each linked pair is set equal to the first. For such calculations, the red-borders marking cells on the worksheets for each type of calculation straddle only one each of these pairs of representations. This is illustrated in Figure 1.28 for the example of the construction of the direct sum character by combination of the irreducible characters of the point group Csh. 

Although Fq describes a set of equivalent elements, it is not degenerate, since it can be further reduced into invariant subspaces. For the case of a triangle this representation gives rise to two irreducible representations (irreps) of Csv... 

As we see, the complexity of (4) is already reduced by symmetry rules, which allow us to identify the non zero vibronic coupling constants. Moreover, the application of the Wigner-Eckart theorem [33, 81] yields a further reduction of the complexity for degenerate irreducible representations. 

---