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 = 12-100. The complete set of two-electron QED corrections is evaluated to all orders in the parameter αZ. Uncalculated contributions to energy levels come through orders α 3 (αZ) 2 , α 2 (αZ) 7 , and higher. The calculation presented is the first treatment for excited states of He-like ions complete through order α 2 (αZ) 4 . A significant improvement in accuracy of theoretical predictions is achieved, especially in the high-Z region.

A new approach to finite basis sets for the Dirac equation is developed. It solves the problem of spurious states and, as a result, improves the convergence properties of basis set calculations. The 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.