Discontinuities in Kohn-Sham potentials in one and three dimensions

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Abstract
(1) Because one-dimensional Coulomb integrals are infinite, Sham and Schlüter used a model interaction in their calculation of Kohn-Sham exchange potential discontinuity in a one-dimensional semiconductor. We derive a different exchange interaction whose one arbitrary parameter is somewhat better defined than theirs. (2) We derive analytic expressions for the contributions to the Kohn-Sham kinetic-potential discontinuity in one dimension and explain how this discontinuity is a consequence of the continuity of the kinetic energy across the energy gap. (3) We show how improvements to the local-density-approximation exchange potential (in three dimensions) which are incapable of yielding the discontinuity in the exact Kohn-Sham exchange potential may still result in much improved energy gaps.