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

TL;DR

This paper simplifies the optimized effective potential (OEP) method, providing a direct proof, an iterative construction approach, and applying it to atomic and cluster systems to analyze exchange potentials and related properties.

Contribution

It introduces a simplified proof of the OEP equation, an iterative method for constructing the exact exchange potential, and applies these techniques to atomic and cluster systems.

Findings

Exact exchange potential calculated for atoms and sodium clusters.

Long-range behavior of the exchange potential analyzed.

Comparison of OEP with KLI and LDA approximations.

Abstract

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.