Interplay between breathing-mode distortions and magnetic order in rare-earth nickelates from $ab$ $initio$ magnetic models.

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.