Charge distribution and magnetism in bilayer <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>La</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:msub> <mml:mi>Ni</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <

An accurate understanding of the ground state electronic properties of ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$, a high-temperature superconductor under pressure, is key for unveiling the origin of its superconductivity. In this paper, we conduct a theoretical study of the electronic structure of the bilayer polymorph of ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$ using the hybrid functional approach, which is well suited to tackle the nonlocal correlation effects arising in this system from the molecular orbital splitting of the Ni $3{d}_{3{z}^{2}\ensuremath{-}{r}^{2}}$ states inside Ni-Ni dimers. Our calculations reveal that bilayer ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$ is a strongly correlated magnetic system with robust Ni spin moments. Spin moments on individual Ni sites take on unusually small values because of the electron delocalization over molecular orbitals involving multiple Ni and O sites. We further find that the magnetism of bilayer ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$ is intimately linked with charge distribution between different Ni and O orbitals. Two distinct regimes are identified in this regard. In one, molecular orbital physics drives the Ni $3{d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ band towards half-filling, which is a well-established condition for unconventional high-temperature superconductivity upon hole doping in cuprates. In the other, the Ni $3{d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ band is quarter-filled favoring spin- and charge-density wave states and Ni-O bond-disproportionation, which is consistent with several recent experimental claims. It is possible that superconductivity in ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$ occurs as a result of a pressure-induced transition between these two competing regimes. Since none of the low energy phases discovered in this study are metallic, nonstoichiometry would be required for superconductivity to occur.