Hartree electrostatic energy ($E_H=e^2/(8\pi {ϵ}_0)\iint n(\mathbf{r})n({\mathbf{r}}^{\prime })/|\mathbf{r}-{\mathbf{r}}^{\prime }|d^{3}r{d}^3{r}^{\prime }$)

The classical electrostatic self-energy functional of an electron number density is the expression in the title. The factor one-half prevents counting each pair twice. Its functional derivative is the Hartree potential; integrating density times that potential gives $2E_H$. The functional includes an unphysical one-electron self-interaction unless other terms cancel it. It differs from the Hartree energy atomic unit.