Exact exchange plane-wave-pseudopotential calculations for slabs.

The exact exchange of density functional theory is applied to both free-standing graphene and a Si(111) slab, using the plane-wave pseudopotential (PWPP) approach and a periodic repetition of the supercell containing the slab. It is shown that (i) PWPP calculations with exact exchange for slabs in supercell geometry are basically feasible, (ii) the width of the vacuum required for a decoupling of the slabs is only moderately larger than in the case of the local-density approximation, and (iii) the resulting exchange potential vx shows an extended region, both far outside the surface of the slab and far from the middle of the vacuum region between the slabs, in which vx behaves as -e(2)/z, provided the width of the vacuum is chosen sufficiently large. This last result is corroborated by an analytical analysis of periodically repeated jellium slabs. The intermediate -e(2)/z behavior of vx can be used for an absolute normalization of vx and the total Kohn-Sham potential, which, in turn, allows the determination of the work function.