The asymptotic behavior of the exchange potential is derived from the long-range form of the exchange-only Kohn-Sham equations. This is then used to correct the asymptotic potential of a conventional continuum exchange functional. For atomic systems the influence of the correction on the electronic energy can be rigorously determined using a well-known virial relationship; its effect is to make the continuum exchange more like Hartree-Fock exchange. For molecular systems the model nature of the potential leads to problems with translational non-invariance. Molecular energies are therefore not considered, although optimized molecular structures can still be determined, providing internal coordinates are used. The effect of the asymptotic correction is to slightly shorten the bond lengths, again indicating an increased Hartree-Fock character. Possible implications for functional development are considered. (C) 2000 American Institute of Physics. [S0021-9606(00)31308-3].