Intertwined charge and spin instability of La$_3$Ni$_2$O$_7$

Research on nickel-based superconductors has progressed from infinite-layer LaNiO$_2$ to finite-layer La$_{6}$Ni$_{5}$O$_{12}$, and most recently to the Ruddlesden-Popper phase La$_3$Ni$_2$O$_7$, which was found to exhibits onset of superconductivity at $\sim$80\,K under a pressure of $\sim$16\,GPa. Unlike the superconductivity mainly driven by the $d_{x^2-y^2}$ orbital in infinite-layer nickelates, the Ni-$d_{z^2}$ and O-2$p$ orbitals contribute significantly to the low energy states and potentially to the superconducting electron pairing mechanism of La$_3$Ni$_2$O$_7$. Employing density functional calculations and multi-orbital multi-atom cluster exact diagonalization including local exchange and Coulomb interactions, here we analyze the pressure dependent low-energy electronic states of the Ni$_2$O$_9$ cluster, relevant for the bilayer phase of La$_3$Ni$_2$O$_7$. The various possible spin states and the exchange and superexchange mechanisms of the Ni$_2$O$_9$ cluster are quantified via the involvement of the Ni-$3d_{3z^2-r^2}$ orbitals and the atomic Hund's rule exchange, the apical bridging O-$2p_z$ orbitals, and the orbitals involved in the formation of local Zhang-Rice singlet like states. We find that the leading configurations contributiong to the cluster ground-states both for nominal valence and also with local charge fluctuations, do not involve occupation of the apical oxygen, instead they favor formation of in-plane Zhang-Rice singlet like states between an O ligand hole and the Ni $3d_{z^2-y^2}$ orbital. We also highlight two possible charge and spin ordered states suggested by our cluster results, that are nearly degenerate at all relevant pressures within our modelling.