On the relativistic effect in the Dirac--Fock theory

In this paper, we study the error bound of the Dirac--Fock ground-state energy and the Hartree--Fock ground-state energy. This error bound is called the relativistic effect in quantum mechanics. We confirm that the relativistic effect in the Dirac--Fock ground-state energy is of the order $\cO(c^{-2})$ with $c$ being the speed of light. Furthermore, if the potential between electrons and nuclei is regular, we get the leading order relativistic correction, which comprises the sum of the mass-velocity term, the Darwin term, and the spin-orbit term. The proof is based on a delicate study of projections onto the positive eigenspace of some Dirac operators. To our knowledge, it is the first mathematical derivation of the leading order relativistic correction for nonlinear Dirac ground-state energies. Our method paves the way to study the relativistic effects in general nonlinear Dirac problems.