Pseudopotential contributions to the quadrupole moment in charged periodic systems

There has been much interest over many years in studying charged systems after the artificial imposition of periodic boundary conditions, and correcting for resulting divergence electrostatic energy density. A correction cubic cells was derived by Makov Payne 1995, its leading error term is form ${L}^{\ensuremath{-}5}$ a cube side $L$. Most modern density functional theory codes use different treatment $\mathrm{Z}\ensuremath{\alpha}$ to that used Payne, an ${L}^{\ensuremath{-}3}$ if their unmodified. This paper shows how result can be made consistent with practice.