We derive and employ a local potential to represent the Fock exchange operator in electronic single-particle equations. This local Fock-exchange (LFX) potential is very similar to the exact exchange (EXX) potential in density functional theory (DFT). The practical software implementation of the two potentials (LFX and EXX) yields robust and accurate results for a variety of systems (semiconductors, transition metal oxides) where Hartree Fock and popular approximations of DFT typically fail. This includes examples traditionally considered qualitatively inaccessible to calculations that omit correlation.