  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 $E^{\prime}_1$ 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 $\Delta$ 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.
