The electronic structure of the rhombohedral $\mathrm{La}\mathrm{Co}{\mathrm{O}}_3$ perovskite is calculated for a number of structures with varying $\mathrm{Co}{\mathrm{O}}_6$ octahedra geometry. The calculations are based on the generalized gradient approximation (GGA) to the density functional theory and, in order to describe better the strong electron-electron correlation in Co $3d$ states, the $\mathrm{GGA}+\mathrm{U}$ method is employed. Depending on the $\mathrm{Co}-\mathrm{O}$ distance $d$ and the $\mathrm{Co}-\mathrm{O}-\mathrm{Co}$ bond angle $\beta$, the spin $S$ of the cobalt ion is found to correspond either to the intermediate spin state [(IS), $S=1$] or to the low spin state [(LS), $S=0$]. The border line dividing the IS and LS states in the $(d,\beta )$ space is determined. The IS state is more stable than the LS state for structures with longer $\mathrm{Co}-\mathrm{O}$ distances and more open $\mathrm{Co}-\mathrm{O}-\mathrm{Co}$ angles. The $\mathrm{GGA}+\mathrm{U}$ results are in agreement with recently observed pressure-induced IS-LS transition.

Rhodium doping in the LaCo1$-${}xRhxO3 perovskite series (x=0.02\char21{}0.5) has been studied by use of x-ray diffraction, electric transport, and magnetization measurements, complemented by electronic structure GGA+U calculations in a supercell for different concentration regimes. No charge transfer between Co3+ and Rh3+ was evidenced. The diamagnetic ground state of LaCoO3, based on Co3+ in low-spin (LS) state, is disturbed even by a small doping of Rh. The driving force is the elastic energy connected with incorporation of a large Rh3+ cation into the matrix of small LS Co3+ cations, which is relaxed by formation of a large Co3+ in the high-spin (HS) state in the next-nearest sites to the inserted Rh atom. With increasing temperature, the population of Co3+ in HS state increases through thermal excitation, and a saturated phase is obtained close to room temperature, consisting of a nearest-neighbor correlation of small (LS Co3+) and large (HS Co3+ and LS Rh3+) cations in a kind of double perovskite structure. The stabilizing role of the elastic and electronic energy contributions is demonstrated by the supercell calculations for dilute Rh concentration compared to other dopants with various trivalent ionic radius.

${\mathrm{Ba}}_{0.5}{\mathrm{Sr}}_{1.5}{\mathrm{Zn}}_2{\mathrm{Fe}}_{12}{\mathrm{O}}_{22}$ is a promising multiferroic compound in which the electric polarization is intimately connected to the magnetic state. In principle, ferroelectrity might exist above the room temperature, but the electrical conductivity that increases with increasing temperature limits it to temperatures below $\approx 130\mathrm{K}$. We present results of an ab initio electronic structure calculation of the $\left(\mathrm{BaSr}\right){\mathrm{Zn}}_2{\mathrm{Fe}}_{12}{\mathrm{O}}_{22}$ system. To improve the description of strongly correlated $3d$ electrons of iron, the $\mathrm{GGA}+U$ method is used. The results show that the electrical conductivity strongly depends on relative fractions of iron and zinc in the tetrahedral sublattice that belongs to the spinel block of the hexaferrite structure. If this sublattice is fully occupied by zinc, the system is an insulator with a gap of $\approx 1.5\mathrm{eV}$. If it is occupied equally by Fe and Zn the gap decreases by a factor of 2, and the system is metallic when this sublattice is filled by iron only.