  We develop relativistic short-range exchange energy functionals for
four-component relativistic range-separated density-functional theory using a
Dirac-Coulomb Hamiltonian in the no-pair approximation. We show how to improve
the short-range local-density approximation exchange functional for large
range-separation parameters by using the on-top exchange pair density as a new
variable. We also develop a relativistic short-range generalized-gradient
approximation exchange functional which further increases the accuracy for
small range-separation parameters. Tests on the helium, beryllium, neon, and
argon isoelectronic series up to high nuclear charges show that this latter
functional gives exchange energies with a maximal relative percentage error of
3 %. The development of this exchange functional represents a step forward for
the application of four-component relativistic range-separated
density-functional theory to chemical compounds with heavy elements.
