Approximate recalculation of the α(Zα)5 contribution to the self-energy effect on hydrogenic states with a multipole expansion

A contribution of virtual electron states with large wave numbers to the self-energy of an electron bound in the weak Coulomb field is analyzed in the context of the evaluation method suggested in the previous paper. The contribution is of the order α ( Z α ) 5 . The same value for this contribution is found here as the one found in the previous calculations using different evaluation methods. When we add the remaining terms of the order α ( Z α ) 5 to the calculation of the self-energy effect in hydrogen-like ions presented in the previous paper we find a very good agreement with numerical evaluations. The relative difference between present and numerical evaluations ranges from 2 parts in 10 6 for Z = 1 up to 6 parts in 10 4 for Z = 10 .