A Projector Augmented Wave Formulation of the Optimized Effective Potential Formalism

The optimized effective potential (OEP) or exact exchange (EXX) formalism has recently received renewed attention\footnote{ S. K\"{u}mmel and L. Kronik, RMP {\bf{80}}, 3 (2008).} as a method which can improve the accuracy of density functional calculations by representing orbital-dependent functionals and avoiding self-interaction errors found in density functionals. Since the Projector Augmented Wave (PAW) formalism\footnote{ P. Bl\"{o}chl, PRB {\bf{50}}, 17953 (1994); N. A. W. Holzwarth {\em{et al}}, PRB {\bf{55}}, 2005 (1997).} ensures accurate evaluation of interaction integrals by controlling the multipole moments,\footnote{ J. Paier {\em{et al}}, JCP {\bf{122}}, 234102 (2005).} it is a natural choice for implementing OEP within an efficient pseudopotential-like scheme. We developed a frozen core approximation scheme for the atomic all-electron OEP formalism, partitioning the exchange potential into core and valence contributions. The corresponding valence exchange pseudopotential for PAW, $\widetilde{V}_x^{\rm{vale}}({\bf{r}})$, can be derived in a similar way so that for $r > R_c$, $\widetilde{V}_x^{\rm{vale}}({\bf{r}})= {V}_x^{\rm{vale}}({\bf{r}})$. We have investigated the behavior of PAW-OEP basis, projector, and pseudopotential functions for several elements throughout the periodic table.