  The Bogoliubov-Dirac-Fock (BDF) model is a no-photon approximation of quantum
electrodynamics. It allows to study relativistic electrons in interaction with
the Dirac sea. A state is fully characterized by its one-body density matrix,
an infinite rank nonnegative projector. We prove the existence of the
para-positronium, the bound state of an electron and a positron with
antiparallel spins, in the BDF model represented by a critical point of the
energy functional in the absence of external field. We also prove the existence
of the dipositronium, a molecule made of two electrons and two positrons that
also appears as a critical point. More generally, for any half integer $j\in
\tfrac{1}{2}+\mathbb{Z}_+$, we prove the existence of a critical point of the
energy functional made of $2j+1$ electrons and $2j+1$ positrons.
