Symmetry of crystalline orbitals

By consideration of the hexagonal and trigonal structures we conclude the discussion of the structure definitions by space groups and Wyckoff positions. In the next chapter we consider the symmetry of crystalline orbitals, both canonical and localized. 

Itanslation and Space Symmetry of Crystalline Orbitals. Bloch Functions... 

When the space group is realized in a crystalline structure the atomic states included in the LCAO basis define the symmetry of crystalline orbitals appearing in the electronic-structure calculations. The symmetry connection of atomic and crystalline orbitals is given by induced representations of space groups considered in the next subsection. 

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. 

Symmetry Adapted Crystalline Orbitals in SCF-LCAO Periodic Calculations. I. The Construction of the Symmetrized Orbitals. 

Due to the considered symmetry difference of crystalline orbitals in MgO and Si crystals the nature of chemical bonding in these crystals is also different. Indeed, in ionic MgO crystal the splitting of the valence band allows the crystalline orbitals locahzed on an oxygen atom to be generated and transformed over aig and irreducible representations of the oxygen site-symmetry group O. In covalent Si crystal aU four sheets of the valence band have to be included in localization so that the locahzed orbitals found are centered at the middle of the Si — Si bond. 

So far, we have discussed the crystalline field acting on the ion A due to an octahedral environment of six B ligand ions. In many optically ion activated crystals, such as Ti +rAlaOj, the local symmetry of the active ion A is slightly distorted from the perfect octahedral symmetry Oh symmetry). This distortion can be considered as a perturbation of the main octahedral field. In general, this perturbation lifts the orbital degeneracy of the tag and eg levels and then produces additional structure in the tag eg absorption/emission bands. 

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