  A variational solution procedure is reported for the many-particle no-pair
Dirac-Coulomb-Breit Hamiltonian aiming at a parts-per-billion (ppb) convergence
of the atomic and molecular energies, described within the fixed nuclei
approximation. The procedure is tested for nuclear charge numbers from $Z=1$
(hydrogen) to $28$ (iron). Already for the lowest $Z$ values, a significant
difference is observed from leading-order Foldy-Woythusen perturbation theory,
but the observed deviations are smaller than the estimated self-energy and
vacuum polarization corrections.
