Electron-lattice interactions strongly renormalize the charge transfer energy in the spin-chain cuprate Li$_2$CuO$_2$

Strongly correlated insulators are broadly divided into two classes: Mott-Hubbard insulators, where the insulating gap is driven by the Coulomb repulsion $U$ on the transition-metal cation, and charge-transfer insulators, where the gap is driven by the charge transfer energy $\Delta$ between the cation and the ligand anions. The relative magnitudes of $U$ and $\Delta$ determine which class a material belongs to, and subsequently the nature of its low-energy excitations. These energy scales are typically understood through the local chemistry of the active ions. Here we show that the situation is more complex in the low-dimensional charge transfer insulator Li$_\mathrm{2}$CuO$_\mathrm{2}$, where $\Delta$ has a large non-electronic component. Combining resonant inelastic x-ray scattering with detailed modeling, we determine how the elementary lattice, charge, spin, and orbital excitations are entangled in this material. This results in a large lattice-driven renormalization of $\Delta$, which significantly reshapes the fundamental electronic properties of Li$_\mathrm{2}$CuO$_\mathrm{2}$.