Projector Augmented Wave formulation of orbital-dependent exchange-correlation functionals

The use of orbital-dependent exchange-correlation functionals within electronic structure calculations has recently received renewed attention for improving the accuracy of the calculations, especially correcting self-interaction errors. Since the Projector Augmented Wave (PAW) method\footnote{ P. Bl\"{o}chl, {\em{Phys. Rev. B}} {\bf{50}}, 17953 (1994).} is an efficient pseudopotential-like scheme which ensures accurate evaluation of all multipole moments of direct and exchange Coulomb integrals, it is a natural choice for implementing orbital-dependent formalisms. Using Fock exchange as an example of an orbital-dependent functional, we developed the formulation and numerical implementation of the approximate optimized effective potential formalism of Kreiger, Li, and Iafrate (KLI)\footnote{ J. B. Krieger, Y. Li, and G. J. Iafrate {\em{Phys. Rev. A}} {\bf{45}}, 101 (1992).} within the PAW method, comparing results with the analogous Hartree-Fock treatment.\footnote{ Xiao Xu and N. A. W. Holzwarth, {\em{Phys. Rev. B}} {\bf{81}}, 245105 (2010); {\bf{84}}, 155113 (2011).} Test results are presented for ground state properties of two well-known materials -- diamond and LiF. This formalism can be extended to treat orbital-dependent functionals more generally.