The crucial importance of the $t_{2g}$--$e_g$ hybridization in transition metal oxydes

We studied the influence of the trigonal distortion of the regular octahedron along the (111) direction, found in the $\rm CoO_2$ layers. Under such a distortion the $t_{2g}$ orbitals split into one $a_{1g}$ and two degenerated $e_g^\prime$ orbitals. We focused on the relative order of these orbitals. Using quantum chemical calculations of embedded clusters at different levels of theory, we analyzed the influence of the different effects not taken into account in the crystalline field theory; that is metal-ligand hybridization, long-range crystalline field, screening effects and orbital relaxation. We found that none of them are responsible for the relative order of the $t_{2g}$ orbitals. In fact, the trigonal distortion allows a mixing of the $t_{2g}$ and $e_g$ orbitals of the metallic atom. This hybridization is at the origin of the $a_{1g}$--$e_g^\prime$ relative order and of the incorrect prediction of the crystalline field theory.