  We use density-functional theory calculations to explore the magnetic
properties of perovskite rare-earth nickelates, $\mathcal{R}$NiO$_3$, by
constructing microscopic magnetic models containing all relevant exchange
interactions via Wannierization and Green's function techniques. These models
elucidate the mechanism behind the formation of antiferromagnetic order with
the experimentally observed propagation vector, and explain the reason previous
DFT plus Hubbard $U$ calculations favored ferromagnetic order. We perform
calculations of magnetic moments and exchange-coupling parameters for different
amplitudes of the $R_1^+$ breathing mode distortion, which results in expanded
and compressed NiO$_6$ octahedra. We find that the magnetic moment vanishes for
the "short bond" nickels, i.e., the ones in the compressed octahedra. The
inclusion of spin-orbit coupling demonstrates that the magnetic anisotropy is
very small, while the magnetic moment of the short bond nickel atoms tend to
zero even for the noncollinear case. Our results provide a clear picture of the
trends of the magnetic order across the nickelate series and give insights into
the coupling between magnetic order and structural distortions.
