In density-functional theory (DFT), Perdew, Parr, Levy, and Balduz [Phys. Rev. Lett. 49, 1691 (1982)] have shown that for all the electronic systems, the energy of the highest occupied molecular orbital (HOMO) is equal to the negative of the ionization potential. This equality is not recovered within the different approximations of the exchange-correlation functional proposed in the literature. The main reason is that the exchange-correlation potentials of various functionals used in DFT calculations decay rapidly to zero whereas they should exhibit a Coulombic asymptotic -1/r behavior. In this work we propose a gradient-corrected (GC) exchange potential with a correct asymptotic -1/r form for large values of r. The energy of the HOMO calculated with this potential is improved compared to the local-density approximation (LDA) and to the GC functionals widely used in the DFT. Our HOMO eigenvalues are compared to the optimized-potential-model eigenvalues which are the exact values for the exchange-only potential. Using the fact that the LDA satisfies the virial theorem, the exchange energy corresponding to this GC exchange potential can be calculated under a simple assumption.