t 2 g d orbital ordering patterns in KBF 3 (B = Sc, Ti, Fe, Co) perovskites

The orbital ordering (OO) resulting from the partial occupancy of the $t_{2g}$ $d$ subshell of the transition metals in KBF${}_3$ (B = Sc, Ti, Ffe, Co) perovskites, and the many possible patterns arising from the coupling between the B sites, have been investigated at the quantum mechanical level ($\text{all electron}$ Gaussian type basis set, B3LYP hybrid functional) in a 40 atoms supercell. The numerous patterns are distributed into 162 classes of equivalent configurations. For each fluoroperovskite, one representative per class has been calculated. The four compounds behave similarly: an identical dependence of the energy and volume (or cell parameters) on the OO pattern; the spanned energy interval is small (1 to 2 mE${}_h$ per formula unit), suggesting that most of the configurations are occupied at room and even at low temperature. A linear model, taking into account the relative orbital order in contiguous sites, reproduces the energy order in the full set for each compound, suggesting that it could be used for studying OO in larger supercells.