Kohn-Sham energy double-counting correction ($E={\sum }_i{f}_i{ϵ}_i-E_H+E_{\mathrm{xc}}-\int n{v}_{\mathrm{xc}}$)

Summing the Kohn-Sham equations over occupied normalized orbitals yields ${\sum }_i{f}_i{ϵ}_i=T_s+\int nv+2E_H+\int n{v}_{\mathrm{xc}}$. The exact ground-state energy functional is $T_s+\int nv+E_H+E_{\mathrm{xc}}$, so subtraction proves the correction in the title. Nuclear or lattice source self-energies must be added consistently. This accounting does not turn all auxiliary orbital eigenvalues into physical excitation energies.