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The self-consistent one-electron operator obtained by varying a Slater determinant. It contains a local hartree potential and a nonlocal exchange action. Its occupied eigenfunctions define a stationary determinant; its eigenvalues are Lagrange multipliers and are not by themselves the total many-electron energy.

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.