(For context, see my other question here; this has been disproven (albeit contestedly), and is not even well-defined in the first place, so I'm going to ask a slightly different (and well-defined) question here, in a way that wzkchem5's ctrxmpl. might not hold* here)
Let A be a singlet molecule with a non-degenerate ground state and a non-degenerate first excited state. Let X be the set of exact restricted Kohn-Sham orbitals of the ground state of A. The sum of the squares of the "occupied" parts of X equals the exact density of the ground state of A.
Suppose one does the same "summing" process again, except for α-orbital of the HOMO which is swapped with the β-orbital of the LUMO(w.l.o.g, since the α and β orbitals corresponding to an element of X must be the same in their orbit part anyway).
Would this process yield the exact density of the first excited state(non-degenerate by assumption)?
*If one were to apply the finite truncations of wzkchem5's ctrxmpl. to my question and let the distance go to infinity, the "first excited state" might have a chance to converge to the ground state of the limit of the wavefunctions, instead of the first excited state of said limit. I'm a mathematics student so I know what I'm talking about.