Electronic structure and magnetic properties of La$_{3}$Ni$_{2}$O$_{7}$ under pressure: active role of the Ni-$d_{x^2-y^2}$ orbitals

Following the recent report of superconductivity in the bilayer nickelate La$_{3}$Ni$_{2}$O$_{7}$ under pressure, we present an analysis of the electronic and magnetic properties of La$_{3}$Ni$_{2}$O$_{7}$ as a function of pressure using correlated density functional theory methods (DFT+$U$). At the bare DFT level, the electronic structure of the ambient and high-pressure phases of La$_{3}$Ni$_{2}$O$_{7}$are qualitatively similar. Upon including local correlation effects within DFT+$U$ and allowing for magnetic ordering, we find a delicate interplay between pressure and electronic correlations. Within the pressure-correlations phase space, we identify a region (at $U$ values consistent with constrained RPA) characterized by a high spin to low spin transition with increasing pressure. In contrast to previous theoretical work that only highlights the crucial role of the Ni-$d_{z^2}$ orbitals in this material, we find that the Ni-$d_{x^{2}-y^{2}}$ orbitals are active upon pressure and drive this rich magnetic landscape. This picture is preserved in the presence of oxygen deficiencies.