  Arguments showing that exchange-only optimized effective potential (xOEP)
methods, with finite basis sets, cannot in general yield the Hartree-Fock (HF)
ground state energy, but a higher one, are given. While the orbital products of
a complete basis are linearly dependent, the HF ground state energy can only be
obtained via a basis set xOEP scheme in the special case that all products of
occupied and unoccupied orbitals emerging from the employed orbital basis set
are linearly independent from each other. In this case, however, exchange
potentials leading to the HF ground state energy exhibit unphysical
oscillations and do not represent a Kohn-Sham (KS) exchange potential. These
findings solve the seemingly paradoxical results of Staroverov, Scuseria and
Davidson that certain finite basis set xOEP calculations lead to the HF ground
state energy despite the fact that within a real space (or complete basis)
representation the xOEP ground state energy is always higher than the HF
energy. Moreover, whether or not the occupied and unoccupied orbital products
are linearly independent, it is shown that basis set xOEP methods only
represent exact exchange-only (EXX) KS methods, i.e., proper density-functional
methods, if the orbital basis set and the auxiliary basis set representing the
exchange potential are balanced to each other, i.e., if the orbital basis is
comprehensive enough for a given auxiliary basis. Otherwise xOEP methods do not
represent EXX KS methods and yield unphysical exchange potentials.
