Relativistic Breit operator and its matrix elements

Let us consider in a similar way the relativistic atomic Hamiltonian in Breit approximation (2.1). Terms Hl, H2 and H3 are one-electron operators. It is very easy to find their irreducible forms. Operators H2 and H3 are complete scalars and do not require any transformations, whereas for H we have to substitute (19.1) and (19.2) in (2.2). This gives 

The energy operator of electrostatic interaction has the same expression as in the non-relativistic case (H = Q, where Q is defined by (1.15)). Its irreducible form is given by (19.6). In order to find the irreducible form of the operator of magnetic interactions H-, we make use of expansion (19.6) and then transform the coupling scheme of tensors to one in which the operators acting on one and the same coordinates, would be directly coupled into a tensorial product. This gives finally 

It is evident that the corresponding expression for operator H5 will differ from (19.66) by the multiplier 1/2. 

The most complicated issue is the transformation of the operator of retarding interactions Hr to the irreducible form. The final result is as follows  

Let us find expressions for the matrix elements of these operators for a subshell of equivalent electrons with respect to the relativistic wave functions, in a one-electron case defined by (2.15), and for a subshell, by (9.8). 

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