Coulomb Energy Density Functionals for Nuclear Systems*

The Coulomb exchange and correlation contributions to the energy density functionals were tested in the context of atomic nuclei \footnote{T. Naito, R. Akashi, and H. Liang. PRC~97,~044319~(2018).}. Both the local density approximation (LDA) and generalized gradient approximation (GGA) functionals were investigated. We employed the experimental nuclear charge density as input. The GGA exchange functionals of electron systems can be applied with practical accuracy to atomic nuclei. In contrast, the correlation functionals of electron systems are not applicable for atomic nuclei. Self-consistent calculation of the GGA exchange functional was also tested \footnote{T. Naito, X. Roca-Maza, G. Col\`{o}, and H. Liang. In Progress.}. In most cases, the GGA exchange functional represents the exact-Fock Coulomb energy while one of the GGA-functional coefficients is changed. This fact is remarkable since the numerical cost of GGA is $O(N^3)$, whereas that cost of exact Hartree-Fock approximation is $O(N^4)$ for the self-consistent calculations.