The many-electron Dirac-Coulomb Hamiltonian ${\mathrm{H}}_{\mathrm{dc}}$, on which most calculations of relativistic effects in many-electron atoms are based, has no normalizable eigenfunctions corresponding to atomic bound states. Two alternative Hamiltonians ${\mathrm{H}}_{+}$ and $h_{+}$, which are derivable within the framework of quantum electrodynamics and hence do not suffer from this defect, are considered. They differ from ${\mathrm{H}}_{\mathrm{dc}}$ by the presence of external-field or free positive-energy projection operators in the interaction terms; the Breit operator can be included in ${\mathrm{H}}_{+}$ or $h_{+}$ without any difficulty arising thereby. The use of ${\mathrm{H}}_{+}$ or $h_{+}$ as a starting point for a systematic approach to the calculation of energy levels and transition amplitudes in atomic physics is described. Hartree-Fock (HF) approximations to the eigenfunctions of ${\mathrm{H}}_{+}$ and $h_{+}$ are defined and the related relativistic HF equations are derived. The results are used to clarify the meaning of the solutions of the Dirac-Hartree-Fock (DHF) equations associated with ${\mathrm{H}}_{\mathrm{dc}}$. The reduction of ${\mathrm{H}}_{+}$ and $h_{+}$ to fully equivalent relativistic Schrödinger-Pauli Hamiltonians ${\mathrm{H}}_P^{\mathrm{rel}}$ and $h_P^{\mathrm{rel}}$ is carried out in closed form. The HF equations associated with $h_P^{\mathrm{rel}}$ are found to be simpler than the DHF equations.

• Received 30 November 1979

DOI:https://doi.org/10.1103/PhysRevA.22.348