  We study the reduced Bogoliubov-Dirac-Fock (BDF) energy which allows to
describe relativistic electrons interacting with the Dirac sea, in an external
electrostatic potential. The model can be seen as a mean-field approximation of
Quantum Electrodynamics (QED) where photons and the so-called exchange term are
neglected. A state of the system is described by its one-body density matrix,
an infinite rank self-adjoint operator which is a compact perturbation of the
negative spectral projector of the free Dirac operator (the Dirac sea).
  We study the minimization of the reduced BDF energy under a charge
constraint. We prove the existence of minimizers for a large range of values of
the charge, and any positive value of the coupling constant $\alpha$. Our
result covers neutral and positively charged molecules, provided that the
positive charge is not large enough to create electron-positron pairs. We also
prove that the density of any minimizer is an $L^1$ function and compute the
effective charge of the system, recovering the usual renormalization of charge:
the physical coupling constant is related to $\alpha$ by the formula
$\alpha_{\rm phys}\simeq \alpha(1+2\alpha/(3\pi)\log\Lambda)^{-1}$, where
$\Lambda$ is the ultraviolet cut-off. We eventually prove an estimate on the
highest number of electrons which can be bound by a nucleus of charge $Z$. In
the nonrelativistic limit, we obtain that this number is $\leq 2Z$, recovering
a result of Lieb.
  This work is based on a series of papers by Hainzl, Lewin, Sere and Solovej
on the mean-field approximation of no-photon QED.
