  The optimized effective potential (OEP) is the exact Kohn-Sham potential for
explicitly orbital-dependent energy functionals, e.g., the exact exchange
energy. We give a proof for the OEP equation which does not depend on the chain
rule for functional derivatives and directly yields the equation in its
simplest form: a certain first-order density shift must vanish. This condition
explains why the highest-occupied orbital energies of Hartree-Fock and
exact-exchange OEP are so close. More importantly, we show that the exact OEP
can be constructed iteratively from the first-order shifts of the Kohn-Sham
orbitals, and that these can be calculated easily. The exact exchange potential
$v_\mathrm{x}(\re)$ for spherical atoms and three-dimensional sodium clusters
is calculated. Its long-range asymptotic behavior is investigated, including
the approach of $v_\mathrm{x}(\re)$ to a non-vanishing constant in particular
spatial directions. We calculate total and orbital energies and static electric
dipole polarizabilities for the sodium clusters employing the exact exchange
functional. Exact OEP results are compared to the Krieger-Li-Iafrate (KLI) and
local density approximations.
