On the Proof by Reductio ad Absurdum of the Hohenberg-Kohn Theorem for Ensembles of Fractionally Occupied States of Coulomb Systems

It is demonstrated that the original reductio ad absurdum proof of the generalization of the Hohenberg-Kohn theorem for ensembles of fractionally occupied states for isolated many-electron Coulomb systems with Coulomb-type external potentials by Gross et al. [Phys. Rev. A 37, 2809 (1988)] is self-contradictory since the to-be-refuted assumption (negation) regarding the ensemble one-electron densities and the assumption about the external potentials are logically incompatible to each other due to the Kato electron-nuclear cusp theorem. It is however proved that the Kato theorem itself provides a satisfactory proof of this theorem.