Role of local geometry in the spin and orbital structure of transition metal compounds

We analyze the role of local geometry in the spin and orbital interaction in transition metal compounds with orbital degeneracy. We stress that the tendency observed in the most studied case (transition metals in O-6 octahedra with one common oxygen-common corner of neighboring octahedra-and with similar to 180A degrees metal-oxygen-metal bonds), that ferro-orbital ordering renders antiferro-spin coupling and, vice versa, antiferro-orbitals give ferro-spin ordering, is not valid in the general case, in particular, for octahedra with a common edge and with similar to 90A degrees M-O-M bonds. Special attention is paid to the third case, that of neighboring octahedra with a common face (three common oxygens), which has largely been disregarded until now, although there are many real systems with this geometry. Interestingly enough, the spin-orbit exchange in this case turns out to be simpler and more symmetric than in the first two cases. We also consider, which form the effective exchange takes for different geometries in the case of strong spin-orbit coupling.