[Paper Review] Ionization potentials and electron affinities from the extended Koopmans' theorem in self-consistent Green's function theory

Alicia Rae Welden, Jordan J. Phillips+1 more|arXiv (Cornell University)|May 21, 2015

Advanced Chemical Physics Studies2 references9 citations

TL;DR

This paper proposes using the Extended Koopmans’ Theorem (EKT) applied to the imaginary-time Green’s function in second-order self-energy (GF2) theory to directly compute ionization potentials (IPs) and electron affinities (EAs) without analytic continuation. It demonstrates that EKT with self-consistent GF2 yields systematically underestimated IPs and EAs compared to UCCSD(T) and experimental values, yet enables accurate quasiparticle spectra reconstruction from imaginary-time data with minimal computational cost.

ABSTRACT

One-body Green's function theories implemented on the real frequency axis offer a natural formalism for the unbiased theoretical determination of quasiparticle spectra in molecules and solids. Self-consistent Green's function methods employing the imaginary axis formalism on the other hand can benefit from the iterative implicit resummation of higher order diagrams that are not included when only the first iteration is performed. Unfortunately, the imaginary axis Green's function does not give direct access to the desired quasiparticle spectra, which undermines its utility. To this end we investigate how reliably one can calculate quasiparticle spectra from the Extended Koopmans' Theorem (EKT) applied to the imaginary time Green's function in a second order approximation (GF2). We find that EKT in conjunction with GF2 yields IPs and EAs that systematically underestimate experimental and accurate coupled-cluster reference values for a variety of molecules and atoms. This establishes that the EKT allows one to utilize the computational advantages of an imaginary axis implementation, while still being able to acquire real axis spectral properties. Because the EKT requires negligible computational effort, and can be used with a Green's function from any level of theory, we conclude that it is a potentially very useful tool for the systematic study of quasiparticle spectra in realistic systems.

Motivation & Objective

• To develop a method for directly computing ionization potentials (IPs) and electron affinities (EAs) from the imaginary-time Green’s function without requiring analytic continuation.
• To assess the reliability of the Extended Koopmans’ Theorem (EKT) when applied to self-consistent Green’s function theory on the imaginary axis.
• To determine whether the computational advantages of imaginary-axis Green’s functions can be leveraged to access real-axis quasiparticle spectra.
• To evaluate the performance of EKT with GF2 against high-accuracy UCCSD(T) reference values and experimental data for a range of molecules and atoms.

Proposed method

• Apply the Extended Koopmans’ Theorem (EKT) to the imaginary-time Green’s function G(τ), using the time-derivative of the particle part at τ = β to extract ionization potentials (IPs).
• Use the time-derivative of the hole part of G(τ) near τ = 0+ to extract electron affinities (EAs) directly from the neutral system’s Green’s function.
• Implement the self-consistent second-order Green’s function (GF2) method on the imaginary frequency axis to compute G(τ) with iterative diagrammatic resummation.
• Utilize a Hartree-Fock reference for non-self-consistent G1F2 calculations to compare with self-consistent GF2 results.
• Perform calculations in the atomic orbital basis using the aug-cc-pVDZ basis set, enabling black-box implementation.
• Avoid analytic continuation by directly extracting quasiparticle peaks from the time-derivative of G(τ) at τ = β and τ = 0+.

Experimental results

Research questions

• RQ1 Can the Extended Koopmans’ Theorem (EKT) be used to extract ionization potentials (IPs) and electron affinities (EAs) directly from the imaginary-time Green’s function without analytic continuation?
• RQ2 How accurate are IPs and EAs computed via EKT applied to self-consistent GF2 compared to high-accuracy UCCSD(T) reference values and experimental data?
• RQ3 Does the self-consistent GF2 approach systematically underestimate IPs and EAs compared to non-self-consistent G1F2 or UCCSD(T), and if so, why?
• RQ4 Can the EKT with imaginary-time Green’s functions reliably reconstruct the full quasiparticle spectrum while retaining the computational advantages of the imaginary-axis formalism?

Key findings

• EKT applied to self-consistent GF2 yields ionization potentials (IPs) and electron affinities (EAs) that are systematically underestimated compared to UCCSD(T) reference values and experimental data.
• For the same set of molecules, EAs calculated via EKT on the imaginary-time Green’s function show good agreement with UCCSD(T) values, particularly in weakly correlated systems, though still systematically low.
• Non-self-consistent G1F2 on a Hartree-Fock reference produces slightly larger IPs and EAs than self-consistent GF2, suggesting the underestimation is not solely due to screening effects in GW-like methods.
• The EKT approach allows direct access to quasiparticle spectra from the imaginary-time domain without analytic continuation, preserving computational efficiency.
• The method is computationally inexpensive, applicable to any Green’s function level of theory, and can be implemented in a black-box manner.
• The underestimation of IPs and EAs in GF2-EKT is attributed to insufficient relaxation in the virtual orbital space, particularly for EAs, though the hole part near τ = 0+ remains robust.