All-order calculations of the spectra of Ba II, Ra II, Fr I and superheavy elements E119 I and E120 II

A technique based on summation of dominating classes of correlation diagrams to all orders in Coulomb interaction is used to calculate the energies of the lowest $s,p$ and $d$ states of Ba II, Ra II, Fr I, E119 I and E120 II. Breit and quantum electrodynamic corrections are also included. Comparison with experiment for Ba II, Ra II and Fr I demonstrates that the accuracy of the calculations is on the level of 0.1%. The technique has been applied to predict the spectra of superheavy elements E119 and E120$^+$. The role of the {\em ladder diagrams} (V. A. Dzuba, Phys. Rev. A {\bf 78}, 042502 (2008)) which is the most recent addition to the method has been emphasised. Their inclusion significantly improves the accuracy of the calculations and expands the applicability of the method.