Density-functional-theory gradient expansion approximation for the screened-Coulomb exchange energy

We have extended Sham's work to derive the density-functional-theory (DFT) first-gradient-correction coefficient in the expansion for the screened-Coulomb exchange energy. For finite screening, this coefficient is equivalent to that derived within Hartree-Fock theory (HFT). It reduces to the bare-Coulomb interaction value in the limit of no screening, in which limit, as is well known, the HFT coefficient is singular. Due to a universal feature of the DFT coefficient derived (a feature not possessed by the HFT coefficient), it has been possible to demonstrate in a general manner conclusions on the convergence of this expansion arrived at in previous work of ours.