Exchange Interaction at the Origin

The origin behavior of Xf r)/Ri r) is derived in the following for the non-relativistic and the relativistic Hartree-Fock theory. For the determination of X (0)/P (0), we evaluate products of potential functions Uijv r) and ratios of radial functions Pj(0)/P (0), 

Here we introduced k in and which denote the lowest indices k and m in Eq. (9.168) with non-zero values for the coefficients and bm. Both and titmin are zero in case of a point nucleus. In analogy to the Hartree-Fock case, we therefore obtain 

Obviously, in the case of an external potential of a point nucleus, both expressions are equal, j6 = j6 = j6y. The next task is the determination of the lowest possible values since only = 0 results in regular, nonva- 

In the presence of the external potential of a point nucleus, the exponents Xi are nonintegral, while the allowed values of v are always integers. Therefore, additional cases must be considered in order to determine those which lead to nonvanishing contributions at the origin in Eqs. (9.179) and (9.180), i.e., which yield exponents j6fj 0. 

This shows that all electron-electron interaction potentials Wf (r), Wp (r) (see Eqs. (9.123) and (9.124)), in which contributions with Ik, Kj occur, i.e., all electron-electron interaction potentials except those with the minimal Kj = 1, behave nonregularly at the origin. 

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