Exchange-correlation potentials with proper discontinuities for physically meaningful Kohn-Sham eigenvalues and band structures

Abstract

It is shown how to properly construct exchange-correlation (xc) potentials in the Kohn-Sham (KS) formalism of density-functional theory such that physically meaningful KS eigenvalues result. A potential adjustor ${\mathrm{\ensuremath{\Delta}}}_{Hxc}^{N\ensuremath{-}}$ is derived which enables for any approximate functional for the xc energy the construction of a consistent xc potential leading to an eigenvalue of the energetically highest occupied KS orbital equal to the negative of the ionization potential. Together with a second potential adjustor ${\mathrm{\ensuremath{\Delta}}}_{Hxc}^{N+}$ KS band structures can be converted in approximate quasiparticle band structures exhibiting the exact physical instead of the KS band gap. This represents an alternative route to the fundamental quasiparticle band gap, completely within the KS formalism without the need to resort to many-body perturbation theory approaches like the $GW$ method. It is shown that, for any finite system, approximate xc potentials including those in the local density and generalized gradient approximations, in contrast to common belief, always exhibit consistent nonzero discontinuities at integer electron numbers, if constructed properly. Thus the discontinuity of xc potentials is identified as a derived quantity which emerges automatically by properly constructing xc potentials without the need to be specifically incorporated in approximate xc functionals. It is demonstrated that all relevant objects of the ensemble KS formalism, in particular functional derivatives and their discontinuities, can be expressed in terms of quantities readily available in the KS formalism of integer electron numbers within an approach named integer electron ensemble approach (IEEA). Attempts to specifically construct ensemble density functionals are shown to be needless. Taking the known asymptotic behavior of the electron density into account an internal consistency condition for xc potentials is presented which justifies tuning procedures of xc functionals and, furthermore, indicates how asymptotic corrections for xc potentials have to be properly employed.