Exact exchange potential for slabs: Asymptotic behavior of the Krieger-Li-Iafrate approximation

Abstract

The Krieger-Li-Iafrate (KLI) approximation for the exact exchange (EXX) potential of density functional theory is investigated far outside the surface of slabs. For large $z$ the Slater component of the EXX/KLI potential falls off as $\ensuremath{-}1/z$, where $z$ is the distance to the surface of a slab parallel to the $xy$ plane. The Slater potential thus reproduces the behavior of the exact EXX potential. Here it is demonstrated that the second component of the EXX/KLI potential, often called the orbital-shift term, is also proportional to $1/z$ for large $z$, at least in general. This result is obtained by an analytical evaluation of the Brillouin zone integrals involved, relying on the exponential decay of the states into the vacuum. Several situations need to be distinguished in the Brillouin zone integration, depending on the band structure of the slab. In all standard situations, including such prominent cases as graphene and Si(111) slabs, however, a $1/z$ dependence of the orbital-shift potential is obtained to leading order. The complete EXX/KLI potential therefore does not reproduce the asymptotic behavior of the exact EXX potential.