First-principles and semiempirical calculations forFcenters inKNbO3

The linear muffin-tin orbital method combined with density functional theory (local approximation) and the semiempirical method of the intermediate neglect of the differential overlap (INDO) based on the Hartree-Fock formalism are used for the study of the $F$ centers (O vacancy with two electrons) in cubic and orthorhombic ferroelectric KNbO${}_3$ crystals. Calculations for 39-atom supercells show that the two electrons are considerably delocalized even in the ground state of the defect. Their wave functions extend over the two Nb atoms closest to the O vacancy and over other nearby atoms. Thus, the $F$ center in KNbO${}_3$ resembles electron defects in the partially-covalent SiO${}_2$ crystal (the so-called $E_1^{\prime }$ center) rather than usual $F$ centers in ionic crystals like MgO and alkali halides. This covalency is confirmed by the analysis of the electronic density distribution. Absorption energies were calculated by means of the INDO method using the Δ self-consistent-field scheme after a relaxation of atoms surrounding the $F$ center. For the orthorhombic phase three absorption bands are calculated to lie at 2.72 eV, 3.04 eV, and 3.11 eV. The first one is close to that observed under electron irradiation. For the cubic phase, stable at high temperatures, above 708 K, only the two bands, at 2.73 eV and 2.97 eV, are expected.