Computation of Second-Order Many-Body Corrections in Relativistic Atomic Systems

A novel technique employing $B$ splines is introduced that permits the precise evaluation of multiple sums over excited states occurring in higher-order many-body perturbation theory. As a test of the technique, the accurately known nonrelativistic expression for the second-order energy of helium in the $\frac{1}{Z}$ approximation is first calculated. The corresponding relativistic expression is then evaluated over the entire helium isoelectronic sequence with attention to the proper treatment of negative-energy states.