Double group representation

Case (b). This case includes no single-group representations, and all real one-dimensional, double-group representations. This includes only the following. 

Case (c). This case includes all single- and double-group representations with complex character. 

Dominant-diagonal theorem, 58 Doob, J. L., 171,174,269 Dorodnitzin, A., 388 Double-group representations, 737 Double groups, 727... 

Since a d and SF El have the same irreducible representations, the inner subdirect product group is often referred to as the double group. The irreducible representations ad of SF El ° d are double-group representations. Half of the irreducible representations of SF El a d are identical to those of y of SF. In fact when H(Qeq) involves an even number of electrons all eigenkets of /f(Qeq) in W will be labeled only by the irreducible representations of SF El a i identical to those of SF. For an odd number of electrons, the converse situation arises. 

The IRs of G comprise the vector representations, which are the IRs of G, and new representations called the spinor or double group representations, which correspond to half-integral j. The double group G contains twice as many elements as G but not twice as many classes g, and g,- are in different classes in G except when g,- is a proper or improper BB rotation (that is, a rotation about a binary axis that is normal to another binary axis), in which case g, and gt are in the same class and (gj, (xg,) are necessarily zero in spinor... 

The symmetry properties of diatomic molecules in the double group representation may be found in the work by Malli and Oreg (44). However, the... 

In the above discussion of the electronic structure of the donor levels, the electron spin has been neglected. It has been, however, proven necessary to introduce the spin-orbit coupling to explain the observation of parity-forbidden transitions for donors with relatively deep ls(Ai) ground states. Using the double group representation of Td, it is found (see Table B.4 of appendix B) that the simple representations Ai and E transform into the T6 and Tg double representations, respectively and that T2 transforms into T7 + Eg. Electric-dipole transitions are symmetry-allowed between A (Tg) and the two T2 (Ty) and T2 (r8) levels. 

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