Validity of the Extended Koopmans' Theorem.

Area of Science:

• Quantum Chemistry
• Theoretical Chemistry
• Computational Chemistry

Background:

• The generalized Koopmans' theorem (EKT) estimates ionization potentials (IPs) via a generalized eigenvalue problem.
• A previous proof suggested EKT predicts exact IPs for Coulomb systems, but later studies challenged this.
• Contradictory findings regarding EKT's accuracy necessitate a re-evaluation of its theoretical basis.

Purpose of the Study:

• To resolve the contradiction surrounding the exactness of the generalized Koopmans' theorem.
• To investigate the existence of the lowest eigenvalue in the EKT procedure.
• To clarify the relationship between EKT-derived IPs and exact ionization energies.

Main Methods:

• Analysis of the generalized eigenvalue problem inherent in the EKT.
• Mathematical demonstration of the conditions under which the lowest eigenvalue may not exist.
• Theoretical investigation of the convergence properties of EKT eigenvalues.

Main Results:

• The lowest eigenvalue of the generalized Koopmans' procedure does not generally exist.
• The non-existence of the lowest eigenvalue explains the conflicting results in the literature.
• EKT eigenvalues converge to the exact lowest ionization energy, acting as an accumulation point.

Conclusions:

• The generalized Koopmans' theorem provides accurate approximations, but its lowest eigenvalue is not guaranteed to exist.
• The theoretical framework of EKT needs refinement to account for the potential non-existence of its lowest eigenvalue.
• Despite this, EKT remains a valuable tool for approximating ionization potentials in quantum systems.