Hartree potential

The local mean Coulomb potential generated by the occupied electron density. A separate positive background or external charge contribution must be included where appropriate. Hartree-Fock exchange is nonlocal and removes equal-orbital self-interaction from the total direct-plus-exchange result.

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.