Hardness of molecules and the band gap of solids within the Kohn-Sham formalism: A perturbation-scaling approach.

Abstract

The finite difference expression for the hardness of atoms and molecules, i.e. half the difference between the ionization potential and the electron affinity according to Parr and Pearson [J. Am. Chem. Soc. 105, 7512 (1983)], is expressed here as an explicit functional of the Kohn-Sham orbitals, of the Kohn-Sham eigenvalue differences, of the Coulomb potential, and of certain parts of the exchange-correlation potential. The functional is derived by exploiting the relationship between uniform coordinate scaling of the electron density and a perturbation theory with respect to the electron-electron interaction. The hardness is obtained as a perturbation expansion consisting of terms which each are connected to a specific order of ${\mathit{e}}^{2}$ with e being the electronic charge. This allows one, in principle, to determine the hardness exactly within the Kohn-Sham method or, in actual applications, up to some chosen order in ${\mathit{e}}^{2}$. The actual expansion is displayed through second order. To some extent the results are also valid in the case of band gaps of solids.