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Connection between $GW$ and Extended Coupled Cluster

Johannes Tölle, Marios-Petros Kitsaras, Andreas Irmler, Andreas Grüneis, Pierre-François Loos

TL;DR

The paper establishes a formal bridge between extended coupled cluster (ECC) theory and the $G_0W_0$ approximation by recasting $GW$ as an electron-boson problem and deriving an ECC-based equation-of-motion framework that reproduces the $GW$ supermatrix. It further demonstrates how ECC can host vertex corrections beyond $GW$ through modifications of the Fock-like and coupling blocks, while maintaining a positive semidefinite self-energy. A perturbative derivation yields a linearized $GW$ density matrix inside ECC, enabling static self-energy corrections and a pathway toward self-consistent-like schemes within CC. Preliminary calculations on a 23-molecule benchmark show that ECC-based vertex corrections can significantly improve IP predictions relative to $G_0W_0$, with competitive performance against EOM-IP-CCSD, and point to a versatile route for systematic inclusion of higher-order correlation effects in Green's-function approaches.

Abstract

Coupled-cluster (CC) theory and Green's function many-body perturbation theory (MBPT) have long evolved as distinct yet complementary frameworks for describing electronic correlation. While CC methods employ exponential wavefunction parametrizations that guarantee size extensivity and systematic improvability, Green's function approaches such as the $GW$ approximation describe quasiparticle and optical excitations through diagrammatic resummations. Recent analyses have established a formal correspondence between these frameworks: the $GW$ approximation is equivalent to an equation-of-motion (EOM) treatment of the direct-ring coupled-cluster doubles (drCCD) method. Within this context, the extended CC (ECC) ansatz offers a unified framework connecting CC and MBPT. This formulation bridges CC-based and Green's function-based methods, providing novel avenues for incorporating vertex corrections within a CC framework that keep a positive semi-definite self-energy and lead to potentially systematically improvable Green's function approaches.

Connection between $GW$ and Extended Coupled Cluster

TL;DR

The paper establishes a formal bridge between extended coupled cluster (ECC) theory and the approximation by recasting as an electron-boson problem and deriving an ECC-based equation-of-motion framework that reproduces the supermatrix. It further demonstrates how ECC can host vertex corrections beyond through modifications of the Fock-like and coupling blocks, while maintaining a positive semidefinite self-energy. A perturbative derivation yields a linearized density matrix inside ECC, enabling static self-energy corrections and a pathway toward self-consistent-like schemes within CC. Preliminary calculations on a 23-molecule benchmark show that ECC-based vertex corrections can significantly improve IP predictions relative to , with competitive performance against EOM-IP-CCSD, and point to a versatile route for systematic inclusion of higher-order correlation effects in Green's-function approaches.

Abstract

Coupled-cluster (CC) theory and Green's function many-body perturbation theory (MBPT) have long evolved as distinct yet complementary frameworks for describing electronic correlation. While CC methods employ exponential wavefunction parametrizations that guarantee size extensivity and systematic improvability, Green's function approaches such as the approximation describe quasiparticle and optical excitations through diagrammatic resummations. Recent analyses have established a formal correspondence between these frameworks: the approximation is equivalent to an equation-of-motion (EOM) treatment of the direct-ring coupled-cluster doubles (drCCD) method. Within this context, the extended CC (ECC) ansatz offers a unified framework connecting CC and MBPT. This formulation bridges CC-based and Green's function-based methods, providing novel avenues for incorporating vertex corrections within a CC framework that keep a positive semi-definite self-energy and lead to potentially systematically improvable Green's function approaches.
Paper Structure (12 sections, 66 equations, 2 figures, 2 tables)

This paper contains 12 sections, 66 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The $GW$ Approximation
  3. $GW$ as a electron-boson problem
  4. Extended Coupled Cluster
  5. ECC on the electron-boson Hamiltonian
  6. EOM on the ECC electron-boson Hamiltonian
  7. Equivalence with $G_0W_0$
  8. Beyond-$GW$ vertex corrections from ECC
  9. $GW$ linearized density matrix from ECC perturbation theory
  10. Computational Details
  11. Numerical results
  12. Conclusion

Figures (2)

  • Figure 1: Violin plots of the errors (in ) for the principal IPs obtained with various $G_0W_0$ variants and EOM-IP-CCSD with respect to the TBEs for the benchmark set of 23 molecular systems from Ref. Marie_2024b computed with the aug-cc-pVQZ basis. Note that $\Gamma_\text{V+A}^\text{x} = \Gamma_\text{V}^\text{x} + \Gamma_\text{A}^\text{x}$ and $\Gamma_\text{V+A+CR}^\text{x} = \Gamma_\text{V+A}^\text{x} + \Gamma_\text{CR}^\text{x}$.
  • Figure 2: Violin plots of the errors (in ) for the second IPs obtained with various $G_0W_0$ variants and EOM-IP-CCSD with respect to the TBEs for the benchmark set of 23 molecular systems from Ref. Marie_2024b computed with the aug-cc-pVQZ basis. Note that $\Gamma_\text{V+A}^\text{x} = \Gamma_\text{V}^\text{x} + \Gamma_\text{A}^\text{x}$ and $\Gamma_\text{V+A+CR}^\text{x} = \Gamma_\text{V+A}^\text{x} + \Gamma_\text{CR}^\text{x}$.