Linear-response calculation of the effective Coulomb interaction between closed-shell localized electrons: Cu, Zn, and ZnO

A linear-response method to calculate the effective Coulomb interaction ($U$) between closed-shell localized electrons is suggested and applied to the $3d$ closed-shell systems (Cu, Zn, and ZnO) based on plane-wave basis density-functional theory calculations. Since the closed-shell localized states are far below the Fermi level, large local perturbation potential ($\alpha$) projected to the localized states is applied to induce purposeful density response ($\Delta n$). From the $\alpha$, the perturbation potential cost for density response onset, by which the $\Delta n$ begins to be induced, is removed. The main screening channel for the effective Coulomb interaction is the itinerant electrons deoccupied from the perturbed localized states. The Cu, Zn, and ZnO $3d$ electron binding energies are calculated based on the local density approximation plus $U$ with the $U$ values calculated from the linear-response, which are found to be in good agreement with experiments.