Charge disproportionation without charge transfer in the rare-earth nickelates as a possible mechanism for the metal-insulator transition

We study a model for the metal-insulator (MI) transition in the rare-earth nickelates RNiO$_3$, based upon a negative charge transfer energy and coupling to a rock-salt like lattice distortion of the NiO$_6$ octahedra. Using exact diagonalization and the Hartree-Fock approximation we demonstrate that electrons couple strongly to these distortions. For small distortions the system is metallic, with ground state of predominantly $d^8 \, ligand$ character, where $ligand$ denotes a ligand hole. For sufficiently large distortions ($\delta d_{\rm Ni-O} \sim 0.05 - 0.10$ {\AA}), however, a gap opens at the Fermi energy as the system enters a periodically distorted state alternating along the three crystallographic axes, with $(d^8 \, ligand^2)_{S=0}(d^8)_{S=1}$ character, where $S$ is the total spin. Thus the MI transition may be viewed as being driven by an internal volume ``collapse'' where the NiO$_6$ octahedra with two ligand holes shrink around their central Ni, while the remaining octahedra expand accordingly, resulting in the superstructure observed in x-ray diffraction in the insulating phase. This insulating state is an example of charge ordering achieved without any actual movement of the charge, similar to that reported in a prior DMFT study.