Penetrating standard line broadening impact theory for hydrogen-like ions

The standard line broadening theory uses a second-order perturbative expansion for the relevant S-matrices in the self-energy operator (known as collision operator). Strong collisions were not treatable this way and other methods were developed. In the last few years, however, the importance of penetrating collisions, i.e., collisions for which the long-range dipole approximation is not valid, has been realized. Such collisions are of great importance for nonhydrogenic isolated ion lines as well as for hydrogen lines in high-density plasmas. Although the standard impact theory is no longer applicable in these cases, penetration in fact weakens the interaction and makes perturbation theory more applicable. This fact implies that a new theory, with account for penetration could be developed, to allow wide use of the simplicity of the perturbative solutions and properly account for the very important effects of penetration. Although this has been achieved for hydrogen, extending to the hydrogenic ion case did not previously appear possible. In this work we attempt such an extension that provides, in general, a 2-dimensional quadrature in lieu of a 1-dimensional quadrature as in the hydrogen case.