  We present the first calculation for many-electron atoms complete through
fourth order of many-body perturbation theory. Owing to an overwhelmingly large
number of underlying diagrams, we developed a suite of symbolic algebra tools
to automate derivation and coding. We augment all-order single-double
excitation method with 1,648 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.
