Double hybrid approximate functionals have "unoccupied" Kohn-Sham orbitals in their formulations due to their MP2 component; however, the "exact functional" depends only on the "occupied" K-S orbitals.
Thus the only way double hybrid approximations could approximate the "exact" functional is that the "exact" DFT density be theoretically invariant under MP2, or at least MPn where n tends to infinity.
This would mean that- should one apply MPn to a set of K-S orbitals, let n tend to infinity, and sum the squares of the resulting "MPn occupied orbitals"(like one theoretically does when trying to compute the density from an "unperturbed" K-S set), one should get the same density as the one computed without MPn being applied.
Is it mathematically true?
*I'm assuming exact functional here, not accounting for the fact that its exact form is unknown.