Benchmarking calculations with spectroscopic accuracy of level energies and wavelengths in W LVII - W LXII tungsten ions.

Atomic properties of $n=3$ states of the W$^{56+}$ $-$ W$^{61+}$ ions are systematically investigated through two state-of-the-art methods, namely, the second-order many-body perturbation theory, and the multi-configuration Dirac-Hartree-Fock method combined with the relativistic configuration interaction approach. The contributions of valence-valence and core-valence electron correlations, the Breit interaction, the higher-order retardation correction beyond the Breit interaction through the transverse photon interaction, and the quantum electrodynamical corrections to the excitation energies are studied in detail. The excitation energies and wavelengths obtained with the two methods agree with each other within \approx 0.01 %. The present results achieve spectroscopic accuracy and provide a benchmark test for various applications and other theoretical calculations of W$^{56+}$ $-$ W$^{61+}$ ions. They will assist spectroscopists in their assignment and direct identification of observed lines in complex spectra.