  A self-consistent formulation is proposed to generalize the HF scheme with
the incorporation of screening effects. For this purpose in a first step, an
energy functional is defined by the mean value for the full Hamiltonian, not in
a Slater determinant state, but in the result of the adiabatic connection of
Coulomb plus the nuclear (jellium charge) in the Slater determinant.
Afterwards, the energy functional defining the screening approximation is
defined in a diagrammatic way, by imposing a special "screening" restriction on
the contractions retained in the Wick expansion. The generalized
self-consisting set of equations for the one particle orbitals are written by
imposing the extremum conditions. The scheme is applied to the homogeneous
electron gas. After simplifying the discussion by assuming the screening as
static and that the mean distance between electrons is close to the Bohr
radius, the equations for the electron spectrum and the static screening
properties are solved by iterations. The self-consistent results for the
self-energies dispersion does not show the vanishing density of states at the
Fermi level predicted by the HF self-energy spectrum. In this extreme non
retarded approximation, both, the direct and the exchange potentials are
strongly screened, and the energy is higher that the one given by the usual HF
scheme. However, the inclusion of the retardation in the exact solution and the
sum rules associated to the dielectric response of the problem, can lead to
energy lowering. These effects will be considered in the extension of the work.
