QED Corrections ofO(mc2α7lnα)to the Fine Structure Splittings of Helium and He-like Ions

A reformulation of the external potential Bethe-Salpeter formalism is developed for two-electron atoms. QED and relativistic corrections to energy levels of order ${\alpha }^{7}m{c}^2\ln \alpha$ are derived and expressed in terms of expectation values of nonrelativistic operators. Corrections of order ${\alpha }^{7}m{c}^2$ from exchange diagrams are also found. The total contributions of order ${\alpha }^{7}m{c}^2\ln \alpha$ to the $1s2p{}_{}{}^3{}_{}{}^{}P_J^{}{}_{}{}^{}$ fine structure intervals of helium are $\Delta {\nu }_{01}=82.6\mathrm{}$ kHz and $\Delta {\nu }_{12}=-10.0\mathrm{}$ kHz. Results are given for He-like ions up to $Z=12\mathrm{}$ and compared with experiment.