The emergence of relativistic quantum mechanics has been one of the most remarkable developments in quantum physics over the past decades. Since the early work of Dirac, relativity has always been a part of the quantum physical picture. While non-relativistic theories are highly successful in describing quantum systems where particles move at speeds much slower than the speed of light $c$, relativistic effects can become significant in high-precision calculations. Now it became clear that relativistic effects had an essential influence on a number of physical and chemical properties.

In this talk, we discuss recent advances on the study of relativistic effect in Dirac-Fock ground-state energy. We prove that the error bound between Dirac-Fock ground-state energy and its non-relativistic counterpart (i.e., Hartree—Fock ground-state energy) is of the order $O(c^{-2})$, and, when the potential between electrons and nuclei is regular, we obtain the well-known leading order relativistic correction term (i.e. the sum of the mass-velocity term, the Darwin term and the spin-orbit term).