Application of Coulomb energy density functional for atomic nuclei: Case studies of local density approximation and generalized gradient approximation

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 GGA\@. For the correlation Coulomb energies, it is shown that those functionals of electron systems are not suitable for atomic nuclei.