# Is an "exact" double hybrid density the same as the "exact" DFT density?

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.

• +1. You might find this paper relevant, but it only talks about MPn with n=155 at most (not n=infinity). Perhaps you'd like to ask the question "Does MPn converge to FCI as n -> infinity" in a separate question. But I think this question should probably just stick to MP2 (or MPn with finite 'n') since otherwise it seems that you're asking two different questions: whether it's true for finite n, and whether it's true for infinitely large n. Oct 29, 2022 at 21:58
• As for a more direct answer to the question: wouldn't the exact double-hybrid functional need to be different from the exact ordinary-DFT functional, because the former starts with MP2? You may also be interested in density-corrected DFT, in which a more accurate density is used: mattermodeling.stackexchange.com/a/1260/5 Oct 29, 2022 at 22:04
• I think that the more I think about this question, the more confused I get. What do you mean by "exact DFT density"? Do you mean the "exact physical density"? Why would the "exact double hybrid density" be different? Oct 30, 2022 at 0:12
• @NikeDattani the density one gets if one sums the squares of the occupied K-S orbitals Oct 30, 2022 at 5:36
• I got confused by the word "exact". Oct 30, 2022 at 12:34