Symmetry of Localized Crystalline Orbitals. Wannier Functions

Symmetry of Localized Crystalline Orbitals. Wannier Functions... 

The use of localized orbitals for the cluster calculation is an efficient approach for defective crystals. To connect the perfect crystal localized orbitals and molecular cluster one-electron states the molecular cluster having the shape of a superceU was considered [699]. Such a cluster differs from the cyclic cluster by the absence of PBC introduction for the one-electron states. Evidently, the molecular cluster chosen is neutral and stoichiometric but its point symmetry can be lower than that of the cyclic cluster. Let the locaUzed orthogonal crystalline orbitals (Wannier functions) be defined for the infinite crystal composed of snpercells. The corresponding BZ is L-times reduced (the snpercell is supposed to consist of L primitive unit cells). The Wannier functions W r — Ai) are now introdnced for the supercells with the translation vectors and satisfy the following equation ... 

The theory of induced representations of space groups gives the answer to the question of whether it is possible to generate in the space of states of a given energy band the basis of localized functions The answer to this question allows the symmetry connection between delocalized Bloch-type and localized Wannier-type crystalline orbitals to be obtained. This point is discussed in Sect. 3.3. 

In our examples, aU the induced irreps are simple, excluding the BR corresponding to the 6-sheeted lower valence subband (see Fig. 3.3). This band representation is a composite one as it is formed by two simple band representations d,aig) and (6, t u) induced by 0 2s- and Sr 4pstates, respectively. Analysis of the space symmetry of crystalline orbitals is used to consider the possible centers of localization of chemical bonding in crystals. This task requires the Wannier-function definition and is considered in the next section. 

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