  Two (so-called left and right) variants of N-centered ensemble
density-functional theory (DFT) [Senjean and Fromager, Phys. Rev. A 98, 022513
(2018)] are presented. Unlike the original formulation of the theory, these
variants allow for the description of systems with a fractional electron
number. While conventional DFT for open systems uses only the true electron
density as basic variable, left/right N-centered ensemble DFT relies instead on
(i) a fictitious ensemble density that integrates to a central (integral)
number N of electrons, and (ii) a grand canonical ensemble weight $\alpha$
which is equal to the deviation of the true electron number from N. Within such
a formalism, the infamous derivative discontinuity that appears when crossing
an integral number of electrons is described exactly through the dependence in
$\alpha$ of the left and right N-centered ensemble Hartree-exchange-correlation
density functionals. Incorporating N-centered ensembles into existing
density-functional embedding theories is expected to pave the way towards the
in-principle-exact description of an open fragment by means of a pure-state
N-electron many-body wavefunction. Work is currently in progress in this
direction
