  A realistic evaluation of Coulomb potential has been made for some selected
nuclei using the available model-independent data for the charge density and
the recent development of Coulomb energy-density functional. Within the
Woods-Saxon potential as a nuclear component, we are able to quantify the
differences in proton single-particle energies due to the differences from the
model-independent data of the uniform distribution, the two-parameter Fermi
function, as well as the charge density obtained from a microscopic
Hartree-Fock calculation using the effective Skyrme interaction. The obtained
energy differences are generally small in magnitude, namely about 100~keV or
less if the parameters of the charge density models are appropriately
determined. Considerable larger differences appear when the last occupied state
is 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 evaluating
the Coulomb exchange term within a well-established method because it is not
differentiable at the surface of a nucleus. Traditionally, the missing of
exchange term is corrected for by simply excluding the last-proton contribution
to the direct term. %% We also investigate this approach and find that its
effect is simply an introduction of the factor $(Z-1)/Z$ into the Coulomb
direct term. From medium to heavy nuclei (typically beyond the $sd$ shell) the
resulting proton levels are 300-800~keV higher than those obtained with the
exact Fock term. The result for lighter nuclei tends to be opposite because the
factor $(Z-1)/Z$ decreases rapidly towards the limit of $Z\to 1$. Therefore,
this traditional approach should be avoided for a precision nuclear structure
calculation.
