A rigorous justification of the Mittleman's approach to the Dirac--Fock model

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.