Origin Behavior of Interaction Potentials

To analyze the short-range behavior of the potentials for closed-shell atoms [484], we make use of the series expansion of the radial functions around the origin of the coordinate system. The origin behavior of the Hartree-Fock radial functions can be shown to be regular at the origin. 

According to Eqs. (9.125)-(9.128), we need to evaluate the behavior of the potential functions Uijy r), 

Short-Range Electron-Electron Coulomb Interaction 

However, the short-range behavior of the inhomogeneous part Xf r)/Ri r) (with R being S,P, Q and R, the corresponding radial function) of the electron-electron interaction potentials is different in nonrelativistic and relativistic theory because of the exponents in the series expansions of the radial functions, Eqs. (9.167) and (9.168). This difference has its origin in the structure of the differential equations Eq. (9.120), which are of second order, and Eq. (9.122), which are coupled first-order differential equations. The way in which the structure of the differential equations determines the exponents in the radial function s series expansions has been shown above. In the case of Dirac-Hartree-Fock theory, these exponents additionally depend on the type 

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