By using perturbation theory, an expression is derived for the difference between the Hartree-Fock (HF) and optimized effective potential (OEP) exchange-only ground-state energies. With this expression and a model three-electron system, we provide the first formal proof of a specific example where the HF and OEP exchange-only ground-state energies are different. We also give a constrained-search variational argument that the two energies are generally, nevertheless, expected to be very close, consistent with known numerical results. (C) 2003 American Institute of Physics.