By means of a suitably chosen variational principle, Euler equations are derived which generalize the Hartree-Fock equations in such a way as to include two-particle correlations. For a system of
N fermions these equations involve
N single-particle functions and $\frac{1}{2}N(N\ensuremath{-}1)$ pair correlation functions. The Euler equations are optimal in the sense that a more accurate estimation of the total energy would require the introduction of three-particle correlation functions and the retention of terms nonlinear in the two-particle functions. Although intended primarily for use in the calculation of atomic and molecular electronic structure, the Euler equations are here used to determine the Coulomb correlation energy in the case of the high-density electron gas. The result obtained is shown to have the same leading logarithmic divergence with density as that found previously by Gell-Mann and Brueckner.