Optimized effective potential made simple: Orbital functionals, orbital shifts, and the exact Kohn-Sham exchange potential

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 $v_{\mathrm{x}}\left(\mathbf{r}\right)$ for spherical atoms and three-dimensional sodium clusters is calculated. Its long-range asymptotic behavior is investigated, including the approach of $v_{\mathrm{x}}\left(\mathbf{r}\right)$ to a nonvanishing 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 and local density approximations.