Relation to Breit-Pauli form

The turn-over rule can be used naively, if the integrand has no singularities at the boundaries. In the present case there are singularities at r = 0, then additional boundary terms have to be considered, or - equivalently - all dilferentiations have to be understood in the distribution sense [47], which leads to 5-function terms. 

We have so far assumed that the unperturbed equation and all equations arising in the perturbation expansion are solved exactly. We shall show in section 5 that the derived expressions are also valid, if the equations are not solved exactly, but are replaced by stationarity conditions. We shall therefore renounce on the requirement that = 0 o- Further 

To write p for (p ), as one often finds, is only justified, if one defines the meaning of p. For further reformulation we use 

This term is usually interpreted as the interaction between the spin and the orbital angular momentum of the electron. A careful analysis [7] reveals thet the genuine spin-orbit interaction would be twice (290), and that half of this is compensated by the contribution of the Thomson precession. 

The correction term is usually ignored. However, it is important if o is not a good approximation to its exact counterpart. Calculations in basis expansions converge much faster with extension of the basis size [22], if the correction term is included. Note, that the correction term is spin-independent, it does not affect the spin-orbit term. 

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