Electron self-energy partial wave renormalization

4-2 Electron self-energy partial wave renormalization. 

Another simple method that allows for the SE calculations in heavy atoms is the partial wave remormalization (PWR) approach first proposed in [69], [70]. Within this approach the renormalized expression for the lowest-order SE is presented in the form of the partial wave expansion  

The advantage of the PWR approach is its simplicity, the disadvantage is the numerical cancellation of the divergencies which is always less reliable than the analytic one. The disadvantage is partly avoided in another version of PWR based on the multicommutator representation of both the unrenormalized SE expression and its counterterm [71]. In this method the cancellation of divergencies occurs partly analytically and partly numerically. 

Within this approach the unrenormalized SE correction to the atomic state A is obtained directly from the 5-matrix element for the Feynman graph Fig. la after the integration over both time and frequency variables. This yields  

The evaluation of the integral over w in the complex plane leads to the following expression for the real part of Eq(223)  

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