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This work merges two different types of many-electron ensembles, namely theTheophilou-Gross-Oliveira-Kohn ensembles of ground and neutrally-excitedstates, and the more recent $N$-centered ensembles of neutral and chargedground states. On that basis, an in-principle exact and general, so-calledextended $N$-centered, ensemble density-functional theory of charged andneutral electronic excitations is derived. We revisit in this context theconcept of density-functional derivative discontinuity for neutral excitations,without ever invoking nor using the asymptotic behavior of the ensembleelectronic density. The present mathematical construction fully relies on theweight dependence of the ensemble Hartree-exchange-correlationdensity-functional energy, which makes the theory applicable to lattice modelsand opens new perspectives for the description of gaps in mesoscopic systems.

Neutral electronic excitations and derivative discontinuities: An extended $N$-centered ensemble density functional theory perspective