Development of the local (site) symmetry method in the supercell model of a crystal with impurity

The symmetry aspects of the supercell (extended primitive unit cell) model of the crystal with point defect are considered: Wyckoff positions splitting and the change of the one-electron states classification over $\mathbf{k}$-vector (Brillouin zone folding). The supercell choice has to be made in a such a way that the one-electron states at the valence band top and the conduction band bottom for the perfect crystal are reproduced. Wyckoff positions splitting in the supercell model allows one to consider the defective crystal using different site symmetry point groups and also without use of the point symmetry at all. The site symmetry method allows one to predict the real symmetry of the defective crystal. This is demonstrated in the calculations of Cu impurity in LiCl crystal (the cubic symmetry of the perfect crystal is maintained for defective crystal), Fe impurity in SrTiO$_{3}$ crystal (the point defect lowers the symmetry from cubic to tetragonal) and interstitial iodine atom in CsPbI$_{3}$ crystal (the experimentally found I$_{2}^{-}$-dumbell structure is confirmed only in the calculations without point symmetry restriction).