  We test the Coulomb exchange and correlation energy density functionals of
electron systems for atomic nuclei in the local density approximation (LDA) and
the generalized gradient approximation (GGA). For the exchange Coulomb
energies, it is found that the deviation between the LDA and GGA ranges from
around $ 11 \, \% $ in $ {}^{4} \mathrm{He} $ to around $ 2.2 \, \% $ in $
{}^{208} \mathrm{Pb} $, by taking the Perdew-Burke-Ernzerhof (PBE) functional
as an example of the<EMAIL_ADDRESS>For the correlation Coulomb energies, it is shown
that those functionals of electron systems are not suitable for atomic nuclei.
