Self-consistent solution of Dyson’s equation up to second order for atomic systems

In this paper, the single-particle Green’s function approach is applied to the atomic many-body problem. We present the self-consistent solution of the Dyson equation up to second order in the self-energy for nonrelativistic spin-compensated atoms. This Dyson second-order scheme requires the solution of the Hartree–Fock integro-differential equations as a preliminary step, which is performed in coordinate space (i.e., without an expansion in a basis set). To cope with the huge amount of poles generated in the iterative approach to tackle Dyson’s equation in second order, the BAGEL (BAsis GEnerated by Lanczos) algorithm is employed. The self-consistent scheme is tested on the atomic systems He, Be, Ne, Mg, and Ar with spin-saturated ground state ${}^1{S}_0.$ Predictions of the total binding energy, ionization energy, and single-particle levels are compared with those of other computational schemes [density functional theory, Hartree–Fock (HF), post-HF, and configuration interaction] and with experiment. The correlations included in the Dyson second-order algorithm produce a shift of the Hartree–Fock single-particle energies that allow for a close agreement with experiment.