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A realistic evaluation of Coulomb potential has been made for some selectednuclei using the available model-independent data for the charge density andthe recent development of Coulomb energy-density functional. Within theWoods-Saxon potential as a nuclear component, we are able to quantify thedifferences in proton single-particle energies due to the differences from themodel-independent data of the uniform distribution, the two-parameter Fermifunction, as well as the charge density obtained from a microscopicHartree-Fock calculation using the effective Skyrme interaction. The obtainedenergy differences are generally small in magnitude, namely about 100~keV orless if the parameters of the charge density models are appropriatelydetermined. Considerable larger differences appear when the last occupied stateis highly filled and, at the same time, has a small orbital angular momentum.Sulfur isotopes ($Z=16$) are a perfect example of these nuclei. Unfortunately,despite its simplicity, the uniform distribution cannot be used for evaluatingthe Coulomb exchange term within a well-established method because it is notdifferentiable at the surface of a nucleus. Traditionally, the missing ofexchange term is corrected for by simply excluding the last-proton contributionto the direct term. %% We also investigate this approach and find that itseffect is simply an introduction of the factor $(Z-1)/Z$ into the Coulombdirect term. From medium to heavy nuclei (typically beyond the $sd$ shell) theresulting proton levels are 300-800~keV higher than those obtained with theexact Fock term. The result for lighter nuclei tends to be opposite because thefactor $(Z-1)/Z$ decreases rapidly towards the limit of $Z\to 1$. Therefore,this traditional approach should be avoided for a precision nuclear structurecalculation.

Realistic evaluation of Coulomb potential in spherical nuclei and test of the traditional approach