  The Bogoliubov-Dirac-Fock (BDF) model is the mean-field approximation of
no-photon Quantum Electrodynamics. The present paper is devoted to the study of
the minimization of the BDF energy functional under a charge constraint. An
associated minimizer, if it exists, will usually represent the ground state of
a system of $N$ electrons interacting with the Dirac sea, in an external
electrostatic field generated by one or several fixed nuclei. We prove that
such a minimizer exists when a binding (HVZ-type) condition holds. We also
derive, study and interpret the equation satisfied by such a minimizer.
  Finally, we provide two regimes in which the binding condition is fulfilled,
obtaining the existence of a minimizer in these cases. The first is the weak
coupling regime for which the coupling constant $\alpha$ is small whereas
$\alpha Z$ and the particle number $N$ are fixed. The second is the
non-relativistic regime in which the speed of light tends to infinity (or
equivalently $\alpha$ tends to zero) and $Z$, $N$ are fixed. We also prove that
the electronic solution converges in the non-relativistic limit towards a
Hartree-Fock ground state.
