  We prove a HVZ theorem for a general class of no-pair Hamiltonians describing
an atom or positively charged ion with several electrons in the presence of a
classical external magnetic field. Moreover, we show that there exist
infinitely many eigenvalues below the essential spectrum and that the
corresponding eigenfunctions are exponentially localized. The novelty is that
the electrostatic and magnetic vector potentials as well as a non-local
exchange potential are included in the projection determining the model. As a
main technical tool we derive various commutator estimates involving spectral
projections of Dirac operators with external fields. Our results apply to all
nuclear charges less than or equal to 137.
