Coordinate-space approach to the bound-electron self-energy: Self-Energy   screening calculation
  The self-energy screening correction is evaluated in a model in which the effect of the screening electron is represented as a first-order perturbation of the self energy by an effective potential. The effective potential is the Coulomb potential of the spherically averaged charge density of the screening electron. We evaluate the energy shift due to a $1s_{1/2}$, $2s_{1/2}$, $2p_{1/2}$, or $2p_{3/2}$ electron screening a $1s_{1/2}$, $2s_{1/2}$, $2p_{1/2}$, or $2p_{3/2}$ electron, for nuclear charge Z in the range $5 \le Z\le 92$. A detailed comparison with other calculations is made. 
P. Indelicato and Peter J. Mohr