The non-relativistic expansion of Dirac-Coulomb Hamiltonian up to $α^8$ order

This paper calculates the relativistic corrections for the Dirac-Coulomb system through the method of non-relativistic expansion. By expanding the large and small components of the Dirac wave function and the energy eigenvalues in terms of $\alpha^2$ (where $\alpha$ is the fine-structure constant), we obtain iterative equations for calculating the higher-order relativistic corrections of non-relativistic systems. For a single-electron system, the operator results of the iterative equations are consistent with those in the literature Ref[Zhou et al 2023 J. Phys. B At. Mol. Opt. Phys. {\bf 56} 045001]. Using these iterative equations, we numerically calculate the relativistic corrections up to the order of $\alpha^{20}$ for the hydrogen atom using Slater basis, which converge rapidly to the analytical results of the hydrogen atom. For the two-electron Dirac-Coulomb system, we also present iterative equations for calculating high-order energy corrections, as well as numerical energy corrections of ground state up to the order of $\alpha^8$.