  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.
