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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 basisdensity-functional theory calculations. Since the closed-shell localized statesare far below the Fermi level, large local perturbation potential ($\alpha$)projected to the localized states is applied to induce purposeful densityresponse ($\Delta n$). From the $\alpha$, the perturbation potential cost fordensity response onset, by which the $\Delta n$ begins to be induced, isremoved. The main screening channel for the effective Coulomb interaction isthe itinerant electrons deoccupied from the perturbed localized states. The Cu,Zn, and ZnO $3d$ electron binding energies are calculated based on the localdensity approximation plus $U$ with the $U$ values calculated from thelinear-response, which are found to be in good agreement with experiments.

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