General methods for evaluating matrix elements of singular operators in two-electron systems

Abstract

Due to the recent advances in both theory and experiment for the fine structure of two-electron atomic systems, it is necessary to include quantum electrodynamic (QED) effects through orders α6mc2, α7ln(Zα)mc2, and α7mc2, in order to match the experimental precision. These effects can be expressed in terms of a sum of singular operators. A general scheme is given for the evaluation of a wide range of matrix elements of high-order singular QED operators for two-electron atomic systems in Hylleraas coordinates. The scheme presented here can be applied to triplet states with arbitrary angular momentum. A number of useful expressions for the analytical evaluation of radial integrals are derived. An example is given in calculating the Douglas and Kroll terms, and the numerical values of the reduced matrix elements are presented for the 2 3PJ states of helium.