Foundations of the relativistic theory of many-electron atoms

Abstract

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 Schr\"odinger-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.