Calculations with spectroscopic accuracy for the ground configuration (3d(9)) forbidden transition in Co-like ions

We present systematic and large-scale calculations for the fine-structure energy splitting and transition rate between the 3d(9) D-2(3/2,5/2) levels of Co-like ions with 28 <= Z <= 100. Two different fully relativistic approaches are used, based on the multiconfiguration Dirac-Hartree-Fock (MCDHF) theory and the relativistic many-body-perturbation theory (RMBPT). Especially the former gives results of similar accuracy as experiments for a large range of ions. Our calculations are therefore accurate enough to probe Breit and quantum-electro-dynamic effects. To obtain spectroscopic accuracy, we show that it is important to include deep core-valence correlation, down to and including the n = 2 shell. We estimate that the uncertainties of our wavelengths are within the uncertainty of experiments, i.e., 0.02%. We also show that the frequently used flexible atomic code has an inaccurate treatment of the self-energy (SE) contribution and of the M1-transition properties for lower-Z ions. After correcting for the SE calculation, the resulting RMBPT transition energies are in good agreement with the MCDHF ones, especially for the high-Z end of the Co-like sequence