Electron self-energy potential expansion

For electrons in the multicharged ions or even for the valence electrons in heavy atoms the parameter aZ cannot be considered as small. In case of heavy atoms the reason is that the effective QED potentials of the electron interaction with the nucleus are rather short-range and the interaction occurs when the outer electrons penetrate deeply into the core. Therefore the methods described in Section III are not valid anymore and all-orders in aZ methods axe necessary. 

The method that helped to avoid this difficulty was first introduced in [10] and applied to the evaluation of SB for the A-sheU electrons in the mercury atom. The general idea of the method is the potential expansion of the SE Feynman graph for the bound electron. This expansion is depicted in Fig.ll. The divergency is concentrated in the first two terms of this expansion. These terms are usually called zero - potential and one - potential terms. The third term, so called many - potential term, is finite but the most difficult for numerical evaluations. To avoid the evaluation of AE the authors of jlO] rearranged the three terms of the expansion 

The advantage of Eq(209) is the absence of the cumbersome expression for AE. Moreover, the AE contribution is also absent. The main problem is now the evaluation of the difference (AE — AE ) where the cancellation of the remaining logarithmic ultraviolet divergence should be done numerically. Later the same approach wsis used in [54] for highly charged ions. The finite nucleus was also taken into account in [54]. 

Still the practical implementation of this method is not so simple and for wider applications to electrons moving in non-Coulomb numerically defined potentials as in heavy atoms the more straightforward approach based on the direct use of the potentiaJ expansion can be preferable. 

This is mainly due to the development of numerical methods (B-spline approach [62], space discretization [63]) that allow summations to be performed over the complete Dirac spectrum for arbitrary spherically symmetric potentials. 

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Fig. 11. The expansion of the electron self-energy graph in power of the nuclear potential V. The double solid line denotes an electron in the potential held V, and the ordinary solid line denotes the free electron.

In the Hamiltonian (O Eq. 2.4), the nuclear variables are free and not constant and there are no nuclear kinetic energy operators to dominate the potential operators involving these free nuclear variables. The Hamiltonian thus specified cannot be self-adjoint in the Kato sense. The Hamiltonian can be made self-adjoint by clamping the nuclei because the electronic kinetic energy operators can dominate the potential operators which involve only electronic variables. The Hamiltonian (O Eq. 2.2) is thus a proper one and the solutions O Eq. 2.3 are a complete set. But since the Hamiltonian (O Eq. 2.4) is not self-adjoint it is not at all clear that the hoped for eigensolutions of O Eq. 2.5 form a complete set suitable for the expansion (O Eq. 2.7). 

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