Exact ionization potentials from wavefunction asymptotics: The extended Koopmans' theorem, revisited

A simple explanation is given for the exactness of the extended Koopmans' theorem, (EKT) for computing the removal energy of any many-electron system to the lowest-energy ground state ion of a given symmetry. In particular, by removing the electron from a removal orbital of appropriate symmetry that is concentrated in the asymptotic region, one obtains the exact ionization potential and the exact Dyson orbital for the corresponding state of the ion. It is argued that the EKT is not restricted to many-electron systems but holds for any finite many-body system, provided that the interaction vanishes for increasing interparticle distance. A necessary and sufficient condition for the validity of the EKT for any state (not just the lowest-energy states of a given symmetry) in terms of the third-order reduced density matrix is stated and derived.