A new approach to finite basis sets for the Dirac equation is developed. It does not involve spurious states and improves the convergence properties of basis-set calculations. Efficiency of the method is demonstrated for finite basis sets constructed from $B$ splines by calculating the one-loop self-energy correction for a hydrogenlike ion.

We perform ab initio QED calculations of energy levels for the $n=1$ and $n=2$ states of He-like ions with the nuclear charge in the range $Z=12\text{–}100$. The complete set of two-electron QED corrections is evaluated to all orders in the parameter $\alpha Z$. Uncalculated contributions to energy levels come through orders ${\alpha }^3{\left(\alpha Z\right)}^2$, ${\alpha }^2{\left(\alpha Z\right)}^7$, and higher. A significant improvement in accuracy of theoretical predictions is achieved, especially in the high-$Z$ region.

A model operator approach to calculations of the QED corrections to energy levels in relativistic many-electron atomic systems is developed. The model Lamb shift operator is represented by a sum of local and nonlocal potentials which are defined using the results of ab initio calculations of the diagonal and nondiagonal matrix elements of the one-loop QED operator with H-like wave functions. The model operator can be easily included in any calculations based on the Dirac-Coulomb-Breit Hamiltonian. The efficiency of the method is demonstrated by comparison of the model QED operator results for the Lamb shifts in many-electron atoms and ions with exact QED calculations.

Theoretical energy levels of the n = 1 and n = 2 states of hydrogen-like atoms with the nuclear charge numbers 1 ≤ Z ≤ 110 are tabulated. The tabulation is based on ab initio quantum electrodynamics calculations performed to all orders in the nuclear binding strength parameter Zα, where α is the fine structure constant. Theoretical errors due to various effects are critically examined and estimated.

We present the results of an experimental study of magnetic dipole ($M1$) transitions in highly charged argon ions (Ar X, Ar XI, Ar XIV, Ar XV) in the visible spectral range using an electron beam ion trap. Their wavelengths were determined with, for highly charged ions, unprecedented accuracy up to the sub-ppm level and compared with theoretical calculations. The QED contributions, calculated in this Letter, are found to be 4 orders of magnitude larger than the experimental error and are absolutely indispensable to bring theory and experiment to a good agreement. This method shows great potential for the study of QED effects in relativistic few-electron systems.