Explicit approximate relation between reduced two- and one-particle density matrices

A class of approximation of the two-particle density matrix d₂ resembling the Hartree-Fock dependence on the one-particle density matrix d₁ is suggested. The integral relation between d₂ and d₁ is exactly maintained. An optimal choice is performed by means of the Pauli principle. The smallness of the error when applied to Coulomb systems, is qualitatively discussed. Extended Thomas-Fermi theory, as recently introduced by the authors is shortly outlined for the case of Coulomb interaction.