Projection technique for population analysis of atomic orbitals in crystals

The one-electron density matrix of a crystal in the basis set of localized orbitals is calculated using two variants of the projection technique, namely, the projection of crystal orbitals onto the space of atomic orbitals (technique A) and the projection of atomic functions onto the space of crystal orbitals (technique B). A comparative analysis of the one-electron density matrices thus obtained is carried out, and a simplified version of technique B is proposed to avoid cumbersome calculations with a large number of vacant crystal orbitals. Both techniques are used to calculate the local characteristics of the electronic structure (atomic charges, atomic covalences, bond orders) for a number of crystals (Si, SiC, GaAs, MgO, cubic BN, TiO₂ rutile) in the framework of the density-functional theory within the generalized gradient approximation in the plane wave basis set with the norm-conserving pseudopotentials. It is revealed that both variants of the projection technique lead to close local characteristics of the electronic structure. The local characteristics of the electronic structure of the TiO₂ crystal with a rutile structure are determined by the projection technique and by constructing the Wannier-type atomic functions (WTAF) in the minimal valence basis set in the framework of the variational method with the crystal orbitals calculated in the linear combination of atomic orbitals (LCAO) approximation. It is demonstrated that, although the basis sets used for calculating the crystal orbitals differ significantly (plane waves in the projection technique, LCAO in the WTAF method), the local characteristics of the electronic structure are in good agreement.