Let there be some ground-state N-electron species named A. A Dyson orbital for Q is defined as the overlap between Q and (Q with one electron removed). The extended Koopmans theorem now states that the set of Dyson orbitals mathematically spans the same space as do the set of Natural Orbitals from "standard" correlated theories. Since it talks about the "set" of Dyson orbitals, there must be more than one Dyson orbital(w.l.o.g, since an one-electron system can be solved in closed form anyway). The Dyson HOMO(s) can now be defined as the overlap between (ground-state) A and the most stable electronic state(s) of A+. Everything is normal.
The main ambiguity occurs when trying to define the Dyson HOMO-1(s). The most natural approach would be to define it(them) as the overlap between the most stable electronic state(s) of A+ and A(2+), resp. The paper supports this definition by discussing the "kth ionisation potential", which cannot be determined by the latter method(vide infra).
However, the paper, which I could not fully grasp, also seems to define the Dyson HOMO-1(s) as the overlap between ground-state A and (A with the second electron removed). Not only is this nonintuitive, this leads to a circular argument, as one needs to define what orbitals are in the first place to define what the "second electron" is. Defining the Dyson HOMO-1(s) as the overlap between A and (the second-lowest state(s) of A+) also leads to a problem; the paper explicitly discusses multiconfigurational states, for which the exact determination of "the second-lowest state(s)" itself is not straightforward.
My question now follows- does the paper (vide infra) define Dyson orbitals in the former way or the latter way(s)?