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

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 e² 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 e². The actual expansion is displayed through second order. To some extent the results are also valid in the case of band gaps of solids.