  We perform the self-consistent Skyrme Hartree-Fock calculation with the
Coulomb exchange functional in the form of generalized gradient approximation
(GGA). It is found that the Perdew-Burke-Ernzerhof GGA (PBE-GGA) Coulomb
exchange functional is able to reproduce the exact-Fock energy for the nuclei
in a large region of the nuclear chart with one adjustable parameter. The
remaining error of Coulomb exchange energy by the GGA with respect to the
exact-Fock energy dominantly comes from the functional-driven error.
