First-order self-energy correction in hydrogenlike systems

We present a detailed description of a procedure for the numerical evaluation of the first-order self-energy correction for an arbitrary excited state. An efficient schema of the numerical treatment of the many-potential part of the Dirac-Coulomb Green function improves the speed of the computation considerably. This feature is extremely important for higher-order self-energy calculations. We apply this method to the evaluation of the self-energy correction for the excited states with $|\kappa |<~5$ and $n<~5$ for some high-$Z$ ions.