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.

Figure 1

Number of diagrams grows rapidly with the order of MBPT. Here we show number of topologically distinct Brueckner-Goldstone diagrams for transition amplitudes for univalent atoms. We assume that calculations are carried out in $V^{N-1}$ Hartree-Fock basis to minimize the number of diagrams and we do not count “folded” [6] and normalization diagrams. All-order single-double (SD) excitation method recovers all diagrams through the third order, but misses roughly a half of diagrams in the fourth order. These 1576 missed diagrams and 72 related normalization diagrams are explicitly computed in the present work.