Irrep classification

The basis for the classification and canonical ordering of CSF encountered in the Cl method is chains of possible subgroups. When an irreducible representation (irrep) of a group is restricted to a subgroup the representation splits into a direct sum of irrep of the subgroup. If this subduction process is multiplicity-free (i.e. a given irrep of the subgroup occurs at most once in this... 

In [40] a complete classification of the inequivalent simple induced reps is given and all the cases when the reps induced from the irreps of maximal isotropy subgroups can be eqmvalent to one another are considered. First, equivalent induced reps at the same site q eQ arise. 

To understand the nature of a point defect in a crystal and the degree of its influence on the properties of the crystal matrix it is necessary to relate the local energy levels of the defect to the energy-band structure of the perfect crystal. But these two systems (crystal with defect and perfect crystal) have different symmetries and the classification of electron states is made according to irreps of either a point group (for a crystal with a single defect) or a space group (for a perfect crystal). 

However, if the product spans nondegenerate boson irreps, the symmetry is not well-defined, and it is the real and imaginary parts of the density that have a well-defined symmetry. This is the case even for p = and r =, for the groups in which r(R ) /q. We can of course make the classification of the integrals into the four classes regardless of whether the densities belong to a particular boson irrep, but if we wish to exploit boson symmetry we have to address this issue. 

Classification by time-reversal relations overlaps the symmetry classification. For groups without quaternion irreps, integrals in the classes (// //) and must... 

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... 

The number of rows fs is given by (4.2.2), while the number of columns D m, N, S) is given by (7.6.18). Under a unitary transformation (induced by an orbital basis change) the functions in each row are mixed among themselves and carry an irrep of U(m) while under a permutation of electrons (or orbital indices) the functions in each column are mixed among themselves and carry an irrep Dj of S. The number of functions in the array may be enormous but the classification is simple. 

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