  In this paper, we study the relationship between the Dirac--Fock model and
the electron-positron Hartree--Fock model. We justify the Dirac--Fock model as
a variational approximation of QED when the vacuum polarization is neglected
and when the fine structure constant $\alpha$ is small and the velocity of
light $c$ is large. As a byproduct, we also prove, when $\alpha$ is small or
$c$ is large, the no-unfilled shells theory in the Dirac--Fock theory for atoms
and molecules. The proof is based on some new properties of the Dirac--Fock
model.
