Basis of the theory.

The site symmetry group is defined as a local subgroup of a point ( molecular cluster) or a space ( crystalline solid ) group. Wyckoff positions and crystal structures. Two examples: perovskite CaTiO₃ structure ( symmorphic space group Pm-3m) and anatase structure ( nonsymmorphic space group I4₁/amd). Generation of space group irreducible representations on the Bilbao Crystallographic Server. Subperiodic groups. Relations of the irreducible representations of a group and a subgroup: subduced and induced representations of point and space groups. Correlation tables. The SiteSym program on the Bilbao Crystallographic Server and symmetry relations of localized and extended states.

Practical exercises.

The SiteSym program application for two crystalline structures is studied.

1. Perovskite structure. Wyckoff positions and site symmetry of Ca, Ti and O atoms. Symmetry points in the Brillouin zone of the simple cubic lattice. Symmetry of electron states, generated by s, p and d atomic orbitals of constituent atoms. Symmetry of phonons and phase transitions in SrTiO₃ crystal.
2. Anatase structure. Crystallographic and primitive unit cells. Site symmetry of Ti and O atoms. Symmetry points in the Brillouin zone of the body-centered tetragonal lattice. Symmetry of electron states, generated by s, p and d atomic orbitals of constituent atoms. Symmetry of phonons. Site symmetry in two-periodic (2D) groups. Symmetry of (101) anatase layer and the TiO₂ nanotube construction.

Supplementary Material