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

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 $-1/z$, 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.

Figure 1

Exponent ${\mathit{k}}^2-2{\epsilon }_{\mathit{k}\alpha }$ for the highest occupied band at $\mathit{k}$ throughout the first Brillouin zone (indicated by black lines) for (a) graphene, (b) a 6-layer Si(111) slab, and (c) an 18-layer Si(111) slab. The band energies have been obtained by EXX calculations, utilizing the KLI approximation. The discontinuities in the case of the Si(111) slabs result from different bands being the highest occupied one for different $\mathit{k}$.

Figure 2

Integration region (shaded area) for partially filled band with degenerate minima of ${\gamma }_{\mathit{k}\alpha }$ at the boundary of the integration region.

Figure 3

Exponent ${\mathit{k}}^2-2{\epsilon }_{\mathit{k}\alpha }$ for the highest occupied band at $\mathit{k}$ for a seven-layer Al(100) slab: (a) projection on the $k_x-k_y$ plane in one quarter of the first Brillouin zone and (b) 3D representation of the most relevant region on an enlarged scale. Arrows point at the $\mathit{k}$ values for which ${\mathit{k}}^2-2{\epsilon }_{\mathit{k}\alpha }$ assumes its lowest value (the minima are somewhat masked by the finite resolution of the plot). (b) explicitly demonstrates the discontinuity of ${\mathit{k}}^2-2{\epsilon }_{\mathit{k}\alpha }$. The band energies have been obtained with an EXX calculation, utilizing the KLI approximation.

Figure 4

Complete integration region (gray shaded area) of a partially filled band with a global minimum of ${\gamma }_{\mathit{k}\alpha }$ at $\mathit{q}$ versus the relevant integration region for large $z$: the blue shaded region relevant for $z_2$ gives the same percentage contribution to the total integral as the red shaded region relevant for $z_1<z_2$.