Exact Ionization Potentials from Wavefunction Asymptotics: The Extended Koopmans' Theorem, Revisited

TL;DR

This paper revisits the extended Koopmans' theorem, providing a simple explanation for its exactness in calculating ionization potentials by analyzing wavefunction asymptotics and the conditions under which it holds.

Contribution

It offers a new explanation for the exactness of the extended Koopmans' theorem and generalizes its applicability to any finite many-body system with vanishing interactions at large distances.

Findings

Exact ionization potentials can be obtained from wavefunction asymptotics.

The theorem applies to any finite many-body system with vanishing interactions at large distances.

A necessary and sufficient condition involving the third-order reduced density matrix is derived.

Abstract

A simple explanation is given for the exactness of the extended Koopmans' theorem 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 the interaction vanishes for increasing interparticle distance. A necessary and sufficient condition for the validity of the extended Koopmans' theorem 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.