Simple Iterative Construction of the Optimized Effective Potential for Orbital Functionals, Including Exact Exchange

For exchange-correlation functionals that depend explicitly on the Kohn-Sham orbitals, the potential $V_{\mathrm{x}\mathrm{c}\sigma }(\mathbf{r})$ must be obtained as the solution of the optimized effective potential (OEP) integral equation. This is very demanding and has limited the use of orbital functionals. We demonstrate that instead the OEP can be obtained iteratively by solving the partial differential equations for the orbital shifts that exactify the Krieger-Li-Iafrate approximation. Unoccupied orbitals do not need to be calculated. Accuracy and efficiency of the method are shown for atoms and clusters using the exact-exchange energy. Counterintuitive asymptotic limits of the exact OEP are presented.

Figure 1

Left: $V_{\mathrm{x}}(\mathbf{r})$ for the Ne atom versus radial coordinate $r$ in bohr $(a_0)$. Dashed: KLI. Full: OEP. Right: $S(r)$ at different stages of the iteration. Full: $S(r)$ for KLI potential. Long-dashed, short-dashed, and dotted lines: $S(r)$ after first, second, and third iteration. Note that the magnitude of $S(r)$ is reduced by each iteration and goes to zero.