Orbital Ordering and Magnetic Interactions in BiMnO$_3$
  This work is devoted to the analysis of orbital patterns and related to them interatomic magnetic interactions in centrosymmetric monoclinic structures of BiMnO$_3$, which have been recently determined experimentally. First, we set up an effective lattice fermion model for the manganese 3d bands and derive parameters of this model entirely from first-principles electronic structure calculations. Then, we solve this model in terms of the mean-field Hartree-Fock method and derive parameters of interatomic magnetic interactions between Mn ions. We argue that although nearest-neighbor interactions favors the ferromagnetism, they compete with longer-range antiferromagnetic (AFM) interactions, the existence of which is directly related with the peculiar geometry of the orbital ordering pattern realized in BiMnO$_3$ below 474 K. These AFM interactions favor an AFM phase, which breaks the inversion symmetry. The formation of the AFM phase is assisted by the orbital degrees of freedom, which tend to adjust the nearest-neighbor magnetic interactions in the direction, which further stabilizes this phase. We propose that the multiferroelectric behavior, observed in BiMnO$_3$, may be related with the emergence of the AFM phase under certain conditions. 
I.V. Solovyev and Z.V. Pchelkina