We study a model for the metal-insulator ($M\text{$-${}}I$) transition in the rare-earth-element nickelates $R{\mathrm{NiO}}_{3}$, based upon a negative charge transfer energy and coupling to a rocksaltlike lattice distortion of the ${\mathrm{NiO}}_{6}$ 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 a ground state of predominantly ${d}^{8}L$ character, where $\underset{_}{L}$ denotes a ligand hole. For sufficiently large distortions ($$\delta${}{d}_{\mathrm{Ni}\text{$-${}}\mathrm{O}}$\sim${}0.05\char21{}0.10\text{ }\text{ }\AA{}$), however, a gap opens at the Fermi energy as the system enters a periodically distorted state alternating along the three crystallographic axes, with $({d}^{8}{\underset{_}{L}}^{2}{)}_{S=0}({d}^{8}{)}_{S=1}$ character, where $S$ is the total spin. Thus the $M\text{$-${}}I$ transition may be viewed as being driven by an internal volume ``collapse'' where the ${\mathrm{NiO}}_{6}$ octahedra with two ligand holes shrink around their central Ni, while the remaining octahedra expand accordingly, resulting in the ($1/2$, $1/2$, $1/2$) 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.

The large variations in $T_c$ across the cuprate families is one of the major unsolved puzzles in condensed matter physics and is poorly understood. Although there appears to be a great deal of universality in the cuprates, several orders of magnitude changes in $T_c$ can be achieved through changes in the chemical composition and structure of the unit cell. In this paper we formulate a systematic examination of the variations in electron-phonon coupling to oxygen phonons in the cuprates, incorporating a number of effects arising from several aspects of chemical composition and doping across cuprate families. It is argued that the electron-phonon coupling is a very sensitive probe of the material-dependent variations in chemical structure, affecting the orbital character of the band crossing the Fermi level, the strength of local electric fields arising from structural-induced symmetry breaking, doping-dependent changes in the underlying band structure, and ionicity of the crystal governing the ability of the material to screen $c$-axis perturbations. Using electrostatic Ewald calculations and known experimental structural data, we establish a connection between the material’s maximal $T_c$ at optimal doping and the strength of coupling to $c$-axis modes. We demonstrate that materials with the largest coupling to the out-of-phase bond-buckling $\left(B_{1g}\right)$ oxygen phonon branch also have the largest $T_c$’s. In light of this observation we present model $T_c$ calculations using a two-well model where phonons work in conjunction with a dominant pairing interaction, presumably due to spin fluctuations, indicating how phonons can generate sizeable enhancements to $T_c$ despite the relatively small coupling strengths. Combined, these results can provide a natural framework for understanding the doping and material dependence of $T_c$ across the cuprates.

High resolution resonant inelastic x-ray scattering has been performed to reveal the role of lattice coupling in a family of quasi-1D insulating cuprates, ${\mathrm{Ca}}_{2+5x}{\mathrm{Y}}_{2-5x}{\mathrm{Cu}}_5{\mathrm{O}}_{10}$. Site-dependent low-energy excitations arising from progressive emissions of a 70 meV lattice vibrational mode are resolved for the first time, providing a direct measurement of electron-lattice coupling strength. We show that such electron-lattice coupling causes doping-dependent distortions of the Cu-O-Cu bond angle, which sets the intrachain spin exchange interactions. Our results indicate that the lattice degrees of freedom are fully integrated into the electronic behavior in low-dimensional systems.