Fermion irreps

Odd number of electrons —> Fermion irreps of the double group Even number of electrons —> Boson irreps of the double group... 

Figure 2. Symmetry adapted graphical representation of a (6 spinor, 3 electron) space. The totally symmetric irrep is F,. The spinors transform according to the fermion irreps T4 through Fft.

The notation from Altmann and Herzig (1994) is 1/2 and 1/2 for the fermion irreps. However, we prefer the labels 1/2 and -1/2, which display the connection with the mj quantum number (see appendix D). 

It is important to note that 2-spinors and 4-spinors both transform as the fermion irreps of the double group. For 2-spinors, however, there is no factor of in the... 

Here, the fermion irrep is B1/2, the representation matrices are all real, and we have case 1. 

The fermion irreps are 1/2 and -1/2, some of the representation matrices are complex, and we have case 2. Note that although the irreps are one-dimensional (in accordance with Lagrange s theorem), they are labeled in recognition of the equivalence between them together they form an 1/2 rep. 

A correspondence is set up between the fermion irreps in the higher group and the lower group. 

For a group such as C2 , which has a doubly degenerate fermion irrep, fhe group chain to C2 gives some information, but it does not resolve the division between Ai and A2, and B and 2- We therefore need a more general approach. 

The groups D2h, T>2, and C2v, whose fermion irreps are all real, will be called... 

The groups C2h, C2, and Cs, whose fermion irreps are all complex, will be... 

The advantage of having to deal with only one type of fermion irrep will become... 

The two sets of detenmnants therefore form a basis for the doubly degenerate Kramers pairs of the iV-electron states. For the fermion irreps of the binary groups, the only other element of symmetry that can be exploited is inversion, which may be handled in the same way as in nonrelativistic Cl theory. 

The discussion in this section has focused on the behavior of the various open-shell approaches under the symmetry constraints imposed by the double groups. In addition, it should be noted that the degeneracies due to symmetry in the fermion irreps of the double groups are generally lower than those due to spin and boson symmetry in a nonrelativistic calculation. Hence, the need for open-shell calculations to handle real symmetry degeneracies is in general smaller than in the nonrelativistic case. 

The same considerations also apply to the case of two open shells where the product of the fermion irreps for the open shells belongs to a doubly degenerate boson irrep. In this case the reference is a single determinant, related to its partner by the time-reversal operator. Because there is no symmetry between the open shells, we cannot derive relations between the amplitudes for Kramers partners. 

The capability for relativistic hybridization is present for molecules of other than linear symmetry. The symmetry spinors for groups lower than cubic can be classified in terms of the m.j quantum numbers for spinors at the high symmetry point (or axis). Since the j quantum number is not part of the classification, relativistic hybridization can always take place. For example, in Dih the fermion irreps are e /2, e ji, and es/2, for which the Kramers partners have Tw modb = 1/2, 3/2, 5/2, respectively. The i and j = l + j spinors for a given rtij both belong to the same irrep. In... 

Table D3 Basis functions for fermion irreps in terms of boson irreps for groups that have only one axis of order greater than 2...

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