The discussion that follows focuses on the latter type of ensembles, which we could refer to as "pre-defined" ensembles. Indeed, unlike in thermal DFT, the ensemble weights ξ ξ ξ = (ξ 0 , ξ 1 , . . . , ξ I , . . .) that are assigned to each state within the ensemble will always be known before the ensemble DFT calculation is carried out. They are chosen (in principle, arbitrarily) and fixed. In other words, in the exact theory, the Hohenberg-Kohn theorem is established for a given ensemble or, equivalently, for a given set ξ ξ ξ of weight values. The one-to-one correspondence between local potentials and ensemble densities relies on the extension (from pure ground states to ensembles) of the Rayleigh-Ritz variational principle,