On the relativistic effect in the Dirac--Fock theory

In this paper, we study the error bound between the Dirac--Fock ground-state energy and the Hartree--Fock ground-state energy, a quantity known as the relativistic effect in quantum mechanics. We confirm that the relativistic effect in the Dirac--Fock ground-state energy is of the order $\mathcal{O}(c^{-2})$ with $c$ being the speed of light. Furthermore, if the potential between electrons and nuclei is regular, we get the well-known leading order relativistic correction -- the Breit--Pauli term, which is the sum of the mass-velocity term, the Darwin term, and the spin-orbit term. As a consequence, we also show that the same relativistic effects and leading order relativistic correction also hold in a QED model introduced by Mittleman when the vacuum polarization -- a term of the order $\mathcal{O}(c^{-3})$ -- is neglected. To our knowledge, this is the first time in mathematics that the leading-order relativistic correction has been obtained from nonlinear Dirac ground-state energy problems.