Complete fourth-order relativistic many-body calculations for atoms

We report, to our knowledge, the first relativistic calculation for many-electron atoms complete through the fourth order of many-body perturbation theory. Owing to an overwhelmingly large number of underlying diagrams, the calculations are aided by our suite of symbolic algebra tools. We augment all-order single-double excitation method with 1648 omitted fourth-order diagrams and compute amplitudes of principal transitions in Na. The resulting ab initio relativistic electric dipole amplitudes are in an excellent agreement with 0.05%-accurate experimental values. Analysis of previously unmanageable classes of diagrams provides a useful guide to a design of even more accurate, yet practical, many-body methods.