The optimized effective potential (OEP) is (within a certain linear response approximation) the exact exchange-correlation potential of density-functional theory (DFT). In the program DEMON-DYNARHO, we have implemented the OEP at the exchange-only level without the evaluation of four-center integrals using resolution-of-the-identity techniques. We point out that great care must be taken in treating the asymptotic behavior of the OEP in finite basis set methods. Our results are compared with previous work using OEP-like potentials. Our OEP orbital energies are compared with experimental ionization potentials, Hartree-Fock (HF) orbital energies, and with orbital energies from the local density approximation (LDA) potential, with and without asymptotic correction (AC). We find that OEP orbital energies are a much better approximation to experimental ionization potentials than are HF orbital energies. LDA orbital energies also correlate well with OEP orbital energies, except for a molecule-dependent rigid shift, due to the well-known fact that the LDA potential falls off too rapidly at large distances. The resultant underbinding is largely corrected by the AC-LDA potential whose orbital energies correlate well with OEP orbital energies, with typical differences on the order of 0.5 eV. However larger differences between AC-LDA and OEP orbital energies are also observed, particularly for unoccupied orbitals and the reason for this is discussed. As an illustration of how the OEP might be used in practical calculations, we give an example from time-dependent DFT where use of the OEP instead of the AC-LDA potential leads to significant improvement in a key sigma-->pi(*) excitation energy of ethylene.