Role of local geometry in 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 for the most studied case (transition metals in O${}_6$ octahedra with one common oxygen – common corner of neighboring octahedra and with $\sim 180^{\circ}$ metal–oxygen–metal bonds), that ferro-orbital ordering renders antiferro-spin coupling, and, {\it vice versa}, antiferro-orbitals give ferro-spin ordering, is not valid in general case, in particular for octahedra with common edge and with $\sim {90}^{\circ }$ M–O–M bonds. Special attention is paid to the ``third case’', neighboring octahedra with common face (three common oxygens) – the case practically not considered until now, although there are many real systems with this geometry. Interestingly enough, the spin–orbital exchange in this case turns out to be 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 case of strong spin–orbit coupling.