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

All-order relativistic computations for atoms and molecules using an explicitly correlated Gaussian basis