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Using a Rayleigh-Schr\"odinger perturbation expansion of multiband Hubbard models, we present analytic expressions for the superexchange coupling constants between magnetic transition-metal ions of arbitrary separation in Mott-Hubbard insulators. The only restrictions are (i) all ligand ions be closed shell anions and (ii) all contributing interaction paths be of equal length. For short paths, our results essentially confirm the Goodenough-Kanamori-Anderson rules, yet in general there does not exist any simple rule to predict the sign of the magnetic coupling constants. The most favorable situation for ferromagnetic coupling is found for ions with less than half-filled d shells; the (relative) tendency to ferromagnetic coupling increases with increasing path length. As an application, the magnetic interactions of the Cr compounds Rb₂CrCl₄, CrCl₃, CrBr₃, and CrI₃ are investigated, all of which except CrCl₃ are ferromagnets.
Feldkemper et al. (Wed,) studied this question.