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We show that the energy functional of ensemble Density Functional Theory(DFT) [Perdew et al., Phys. Rev. Lett. 49, 1691 (1982)] in systems withattractive interactions is a convex function of the fractional particle numberN and is given by a series of straight lines joining a subset of ground-stateenergies. As a consequence the exchange-correlation (XC) potential is notdiscontinuous for all N. We highlight the importance of this exact result inthe ensemble-DFT description of the negative-U Anderson model. In the atomiclimit the discontinuity of the XC potential is missing for odd N while forfinite hybridizations the discontinuity at even N is broadened. We demonstratethat the inclusion of these properties in any approximate XC potential iscrucial to reproduce the characteristic signatures of the charge-Kondo effectin the conductance and charge susceptibility.

Missing derivative discontinuity of the exchange-correlation energy for attractive interactions: the charge Kondo effect