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arxiv: 2509.03253 · v2 · pith:KGOXIO6Znew · submitted 2025-09-03 · ⚛️ physics.chem-ph · cond-mat.mtrl-sci· cond-mat.str-el· math-ph· math.MP· nucl-th

Parquet theory for molecular systems: Formalism and static kernel parquet approximation

Antoine Marie , Pierre-Fran\c{c}ois Loos This is my paper

Pith reviewed 2026-05-21 22:39 UTC · model grok-4.3

classification ⚛️ physics.chem-ph cond-mat.mtrl-scicond-mat.str-elmath-phmath.MPnucl-th
keywords parquet theoryGW approximationmolecular systemsionization potentialstwo-body vertexself-energyGreen's functionvertex corrections
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0 comments X

The pith

Parquet theory treats all two-body scattering channels on an equal footing and supplies a practical route beyond the GW approximation for molecular ionization potentials.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents the parquet formalism for molecular systems as a way to move past the GW approximation while avoiding the selective inclusion of vertex corrections. It notes that GW accuracy often relies on a cancellation between different corrections, so unbalanced extensions can reduce performance. Parquet theory instead couples the one-body Green's function, self-energy, and two-body vertex through equations that treat electron-hole, particle-particle, and other channels symmetrically. The authors outline the approximations required to solve these equations for molecules, including the static kernel parquet approximation, and evaluate the resulting principal ionization potentials for small systems.

Core claim

The parquet equations couple the one-body Green's function, the self-energy, and the two-body vertex while placing all two-body scattering channels on an equal footing. When closed with the static kernel parquet approximation, this structure supplies a controlled route beyond GW that can be implemented for molecular systems and tested directly on principal ionization potentials.

What carries the argument

The parquet equations, which couple the one-body Green's function, the self-energy, and the two-body vertex while treating all scattering channels equally.

If this is right

  • The method supplies a single framework that includes vertex corrections across channels without relying on the fortuitous cancellations that stabilize GW.
  • Implementation details are given for molecular systems, allowing direct numerical tests on quasiparticle energies.
  • Accuracy assessments focus on principal ionization potentials, providing a concrete metric for small-molecule benchmarks.
  • The equal-footing treatment avoids channel-specific biases that appear when extending GW or T-matrix approaches separately.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the static-kernel results remain stable for small molecules, the same equations could be applied to excitation spectra or larger systems once computational scaling is addressed.
  • The balanced channel treatment suggests a natural way to compare or combine parquet results with channel-specific methods such as the particle-particle T-matrix.
  • The formalism may transfer to periodic systems, offering a route to test whether equal-footing vertex corrections improve band-gap predictions in solids.

Load-bearing premise

The approximations used to close and solve the parquet equations, including the static kernel choice, preserve enough accuracy for ionization potentials without introducing errors that undermine the equal-footing treatment of channels.

What would settle it

Direct comparison of computed principal ionization potentials for small molecules such as water or ammonia against experimental values or high-level benchmarks; large uncontrolled deviations from reference data would indicate that the approximations fail to deliver the promised balance.

Figures

Figures reproduced from arXiv: 2509.03253 by Antoine Marie, Pierre-Fran\c{c}ois Loos.

Figure 1
Figure 1. Figure 1: FIG. 1. Diagrammatic representation of the full one-body [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Perturbation expansion of the full one-body vertex [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Diagrammatic representation of the Dyson equation, [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The three different types of two-particle reducibility [PITH_FULL_IMAGE:figures/full_fig_p003_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Diagrammatic representation of the three Bethe [PITH_FULL_IMAGE:figures/full_fig_p004_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Diagrammatic representation of the self-energy in [PITH_FULL_IMAGE:figures/full_fig_p004_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Schematic representation of the self-consistent algorithm used in this work to solve the parquet equations. [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Diagrammatic representation of the FLEX self-energy. [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Principal IP of [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Histogram of errors, with respect to FCI, of the principal IP of 8 small molecules in the aug-cc-pVTZ basis computed [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
read the original abstract

The $GW$ approximation has become a method of choice for predicting quasiparticle properties in solids and large molecular systems, owing to its favorable accuracy-cost balance. However, its accuracy is the result of a fortuitous cancellation of vertex corrections in the polarizability and self-energy. Hence, when attempting to go beyond $GW$ through inclusion of vertex corrections, the accuracy can deteriorate if this delicate balance is disrupted. In this work, we explore an alternative route that theoretically goes beyond $GW$: the parquet formalism. Unlike approaches that focus on a single correlation channel, such as the electron-hole channel in $GW$ or the particle-particle channel in $T$-matrix theory, parquet theory treats all two-body scattering channels on an equal footing. We present the formal structure of the parquet equations, which couple the one-body Green's function, the self-energy, and the two-body vertex. We discuss the approximations necessary to solve this set of equations, the advantages and limitations of this approach, outline its implementation for molecular systems, and assess its accuracy for principal ionization potentials of small molecular systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript develops the parquet formalism for molecular systems by presenting the coupled equations involving the one-body Green's function, self-energy, and two-body vertex. It introduces the static kernel parquet approximation (SKPA) to render the equations tractable, outlines the implementation for finite molecular systems, and reports an accuracy assessment for principal ionization potentials of small molecules as an alternative route beyond GW that treats all two-body scattering channels on equal footing.

Significance. If the numerical results and error analysis hold, the work supplies a systematic many-body framework that avoids the channel-specific focus of GW or T-matrix methods while preserving a balanced treatment of vertex corrections. The formal derivation, discussion of necessary approximations, and molecular implementation constitute a useful contribution to electronic-structure theory for systems where GW's fortuitous cancellation may not generalize.

major comments (1)
  1. [Section discussing the static kernel parquet approximation and its implementation] The central claim that SKPA preserves equal-footing treatment of channels while yielding controlled accuracy for molecular IPs rests on the assumption that freezing the kernel to its static limit does not reintroduce uncontrolled frequency-dependent imbalances. The manuscript should demonstrate explicitly (e.g., via comparison of dynamic vs. static contributions in the self-energy or vertex for a benchmark molecule) that residual dynamic effects remain negligible for the reported principal IPs; without such evidence the truncation risks undermining the equal-footing premise for discrete spectra.
minor comments (2)
  1. [Formal structure of the parquet equations] Notation for the two-body vertex and its channel decomposition should be introduced with an explicit equation or diagram in the formalism section to aid readers unfamiliar with parquet theory.
  2. [Numerical results and accuracy assessment] The accuracy assessment would benefit from a table listing computed IPs, reference values, and deviations for each molecule rather than summary statements only.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comment. We address the point below and describe the revisions we will incorporate.

read point-by-point responses

  1. Referee: [Section discussing the static kernel parquet approximation and its implementation] The central claim that SKPA preserves equal-footing treatment of channels while yielding controlled accuracy for molecular IPs rests on the assumption that freezing the kernel to its static limit does not reintroduce uncontrolled frequency-dependent imbalances. The manuscript should demonstrate explicitly (e.g., via comparison of dynamic vs. static contributions in the self-energy or vertex for a benchmark molecule) that residual dynamic effects remain negligible for the reported principal IPs; without such evidence the truncation risks undermining the equal-footing premise for discrete spectra.

    Authors: We agree that an explicit comparison of dynamic and static contributions would provide stronger support for the SKPA and its consistency with equal-footing treatment of channels. The original manuscript motivates the static kernel on the grounds that it renders the parquet equations numerically tractable while retaining a balanced description of all scattering channels, but does not include a direct decomposition of frequency-dependent versus static terms in the self-energy or vertex for the benchmark systems. In the revised version we will add a dedicated subsection (with accompanying figure) that performs this decomposition for at least one representative molecule (e.g., H2O). The analysis will quantify the magnitude of the dynamic corrections to the principal ionization potentials and discuss under which conditions they remain small. This addition will directly address the referee’s concern while preserving the manuscript’s central conclusions. revision: yes

Circularity Check

0 steps flagged

No circularity: standard parquet equations plus independent static-kernel approximation

full rationale

The manuscript presents the parquet equations as the established many-body coupling of Green's function, self-energy and two-body vertex, then introduces the static-kernel truncation as an explicit approximation chosen for computational tractability in molecular systems. No equation is shown to be defined in terms of its own output, no fitted parameter is relabeled as a prediction, and no load-bearing uniqueness theorem or ansatz is imported solely via self-citation. The accuracy assessment for ionization potentials is performed by direct numerical solution and comparison to reference data, keeping the derivation chain independent of the reported numerical results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard many-body perturbation theory axioms and introduces controlled approximations to the parquet equations; no new physical entities are postulated.

axioms (1)
  • standard math Existence and analytic properties of the one- and two-body Green's functions and the irreducible vertex in many-body quantum mechanics.
    Invoked when the parquet equations are stated as coupling the Green's function, self-energy, and two-body vertex.

pith-pipeline@v0.9.0 · 5743 in / 1246 out tokens · 43202 ms · 2026-05-21T22:39:56.061971+00:00 · methodology

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Works this paper leans on

137 extracted references · 137 canonical work pages

  1. [1]

    author author L. Hedin ,\ title title New method for calculating the one-particle Green 's function with application to the electron-gas problem , \ https://doi.org/10.1103/PhysRev.139.A796 journal journal Phys. Rev. \ volume 139 ,\ pages A796 ( year 1965 ) NoStop

    work page doi:10.1103/physrev.139.a796 1965

  2. [2]

    Aryasetiawan \ and\ author O

    author author F. Aryasetiawan \ and\ author O. Gunnarsson ,\ title title The GW method , \ https://doi.org/10.1088/0034-4885/61/3/002 journal journal Rep. Prog. Phys. \ volume 61 ,\ pages 237--312 ( year 1998 ) NoStop

    work page doi:10.1088/0034-4885/61/3/002 1998

  3. [3]

    The GW approximation: content, suc- cesses and limitations.WIREs Computational Molecu- lar Science, 8(3):e1344, 2018.doi:10.1002/wcms.1344

    author author L. Reining ,\ title title The GW approximation: Content, successes and limitations , \ https://doi.org/10.1002/wcms.1344 journal journal WIREs Comput. Mol. Sci. \ volume 8 ,\ pages e1344 ( year 2018 ) NoStop

    work page doi:10.1002/wcms.1344 2018

  4. [4]

    The GW Compendium: A Practical Guide to Theoreti- cal Photoemission Spectroscopy.Frontiers in Chem- istry, 7, 2019

    author author D. Golze , author M. Dvorak ,\ and\ author P. Rinke ,\ title title The GW compendium: A practical guide to theoretical photoemission spectroscopy , \ https://doi.org/10.3389/fchem.2019.00377 journal journal Front. Chem. \ volume 7 ,\ pages 377 ( year 2019 ) NoStop

    work page doi:10.3389/fchem.2019.00377 2019

  5. [5]

    Marie , author A

    author author A. Marie , author A. Ammar ,\ and\ author P.-F. \ Loos ,\ title title The GW approximation: A quantum chemistry perspective , \ in\ https://doi.org/10.1016/bs.aiq.2024.04.001 booktitle Advances in Quantum Chemistry ,\ series Novel Treatments of Strong Correlations , Vol. volume 90 \ ( year 2024 )\ pp.\ pages 157--184 NoStop

    work page doi:10.1016/bs.aiq.2024.04.001 2024

  6. [6]

    Neuhauser , author Y

    author author D. Neuhauser , author Y. Gao , author C. Arntsen , author C. Karshenas , author E. Rabani ,\ and\ author R. Baer ,\ title title Breaking the theoretical scaling limit for predicting quasiparticle energies: The stochastic GW approach , \ https://doi.org/10.1103/PhysRevLett.113.076402 journal journal Phys. Rev. Lett. \ volume 113 ,\ pages 0764...

    work page doi:10.1103/physrevlett.113.076402 2014

  7. [7]

    Govoni \ and\ author G

    author author M. Govoni \ and\ author G. Galli ,\ title title Large scale GW calculations , \ https://doi.org/10.1021/ct500958p journal journal J. Chem. Theory Comput. \ volume 11 ,\ pages 2680--2696 ( year 2015 ) NoStop

    work page doi:10.1021/ct500958p 2015

  8. [8]

    Stochastic GW Calculations for Molecules.Journal of Chemical Theory and Computation, 13(10):4997–5003, 2017.doi:10.1021/acs.jctc.7b00770

    author author V. Vl c ek , author E. Rabani , author D. Neuhauser ,\ and\ author R. Baer ,\ title title Stochastic GW calculations for molecules , \ https://doi.org/10.1021/acs.jctc.7b00770 journal journal J. Chem. Theory Comput. \ volume 13 ,\ pages 4997--5003 ( year 2017 ) NoStop

    work page doi:10.1021/acs.jctc.7b00770 2017

  9. [9]

    Wilhelm , author D

    author author J. Wilhelm , author D. Golze , author L. Talirz , author J. Hutter ,\ and\ author C. A. \ Pignedoli ,\ title title Toward GW calculations on thousands of atoms , \ https://doi.org/10.1021/acs.jpclett.7b02740 journal journal J. Phys. Chem. Lett. \ volume 9 ,\ pages 306--312 ( year 2018 ) NoStop

    work page doi:10.1021/acs.jpclett.7b02740 2018

  10. [10]

    author author M. D. \ Ben , author F. H. \ da Jornada , author A. Canning , author N. Wichmann , author K. Raman , author R. Sasanka , author C. Yang , author S. G. \ Louie ,\ and\ author J. Deslippe ,\ title title Large-scale GW calculations on pre-exascale HPC systems , \ https://doi.org/10.1016/j.cpc.2018.09.003 journal journal Comp. Phys. Comm. \ volu...

    work page doi:10.1016/j.cpc.2018.09.003 2018

  11. [11]

    F \"o rster \ and\ author L

    author author A. F \"o rster \ and\ author L. Visscher ,\ title title Low-order scaling G_0W_0 by pair atomic density fitting , \ https://doi.org/10.1021/acs.jctc.0c00693 journal journal J. Chem. Theory Comput. \ volume 16 ,\ pages 7381--7399 ( year 2020 ) NoStop

    work page doi:10.1021/acs.jctc.0c00693 2020

  12. [12]

    Kaltak \ and\ author G

    author author M. Kaltak \ and\ author G. Kresse ,\ title title Minimax isometry method: A compressive sensing approach for Matsubara summation in many-body perturbation theory , \ https://doi.org/10.1103/PhysRevB.101.205145 journal journal Phys. Rev. B \ volume 101 ,\ pages 205145 ( year 2020 ) NoStop

    work page doi:10.1103/physrevb.101.205145 2020

  13. [13]

    F \"o rster \ and\ author L

    author author A. F \"o rster \ and\ author L. Visscher ,\ title title Low-order scaling quasiparticle self-consistent GW for molecules , \ https://doi.org/10.3389/fchem.2021.736591 journal journal Front. Chem. \ volume 9 ,\ pages 736591 ( year 2021 ) NoStop

    work page doi:10.3389/fchem.2021.736591 2021

  14. [14]

    Duchemin \ and\ author X

    author author I. Duchemin \ and\ author X. Blase ,\ title title Cubic-scaling all-electron GW calculations with a separable density-fitting space--time approach , \ https://doi.org/10.1021/acs.jctc.1c00101 journal journal J. Chem. Theory Comput. \ volume 17 ,\ pages 2383--2393 ( year 2021 ) NoStop

    work page doi:10.1021/acs.jctc.1c00101 2021

  15. [15]

    Low-Scaling GW with Benchmark Accuracy and Appli- cation to Phosphorene Nanosheets.Journal of Chem- ical Theory and Computation, 17(3):1662–1677, 2021

    author author J. Wilhelm , author P. Seewald ,\ and\ author D. Golze ,\ title title Low-scaling GW with benchmark accuracy and application to phosphorene nanosheets , \ https://doi.org/10.1021/acs.jctc.0c01282 journal journal J. Chem. Theory Comput. \ volume 17 ,\ pages 1662--1677 ( year 2021 ) NoStop

    work page doi:10.1021/acs.jctc.0c01282 2021

  16. [16]

    F \"o rster \ and\ author L

    author author A. F \"o rster \ and\ author L. Visscher ,\ title title Quasiparticle self-consistent GW - Bethe--Salpeter equation calculations for large chromophoric systems , \ https://doi.org/10.1021/acs.jctc.2c00531 journal journal J. Chem. Theory Comput. \ volume 18 ,\ pages 6779--6793 ( year 2022 a ) NoStop

    work page doi:10.1021/acs.jctc.2c00531 2022

  17. [17]

    author author V. W.-Z. \ Yu \ and\ author M. Govoni ,\ title title GPU acceleration of large-scale full-frequency GW calculations , \ https://doi.org/10.1021/acs.jctc.2c00241 journal journal J. Chem. Theory Comput. \ volume 18 ,\ pages 4690--4707 ( year 2022 ) NoStop

    work page doi:10.1021/acs.jctc.2c00241 2022

  18. [18]

    T \"o lle , author N

    author author J. T \"o lle , author N. Niemeyer ,\ and\ author J. Neugebauer ,\ title title Accelerating analytic-continuation GW calculations with a Laplace transform and natural auxiliary functions , \ https://doi.org/10.1021/acs.jctc.3c01264 journal journal J. Chem. Theory Comput. \ volume 20 ,\ pages 2022--2032 ( year 2024 ) NoStop

    work page doi:10.1021/acs.jctc.3c01264 2022

  19. [19]

    Strinati , author H

    author author G. Strinati , author H. J. \ Mattausch ,\ and\ author W. Hanke ,\ title title Dynamical correlation effects on the quasiparticle Bloch states of a covalent crystal , \ https://doi.org/10.1103/PhysRevLett.45.290 journal journal Phys. Rev. Lett. \ volume 45 ,\ pages 290--294 ( year 1980 ) NoStop

    work page doi:10.1103/physrevlett.45.290 1980

  20. [20]

    author author M. S. \ Hybertsen \ and\ author S. G. \ Louie ,\ title title First-principles theory of quasiparticles: Calculation of band gaps in semiconductors and insulators , \ https://doi.org/10.1103/PhysRevLett.55.1418 journal journal Phys. Rev. Lett. \ volume 55 ,\ pages 1418--1421 ( year 1985 ) NoStop

    work page doi:10.1103/physrevlett.55.1418 1985

  21. [21]

    author author M. S. \ Hybertsen \ and\ author S. G. \ Louie ,\ title title Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies , \ https://doi.org/10.1103/PhysRevB.34.5390 journal journal Phys. Rev. B \ volume 34 ,\ pages 5390--5413 ( year 1986 ) NoStop

    work page doi:10.1103/physrevb.34.5390 1986

  22. [22]

    author author R. W. \ Godby , author M. Schl\"uter ,\ and\ author L. J. \ Sham ,\ title title Self-energy operators and exchange-correlation potentials in semiconductors , \ https://doi.org/10.1103/PhysRevB.37.10159 journal journal Phys. Rev. B \ volume 37 ,\ pages 10159--10175 ( year 1988 ) NoStop

    work page doi:10.1103/physrevb.37.10159 1988

  23. [23]

    \ Sch \"o ne \ and\ author A

    author author W.-D. \ Sch \"o ne \ and\ author A. G. \ Eguiluz ,\ title title Self-consistent calculations of quasiparticle states in metals and semiconductors , \ https://doi.org/10.1103/PhysRevLett.81.1662 journal journal Phys. Rev. Lett. \ volume 81 ,\ pages 1662--1665 ( year 1998 ) NoStop

    work page doi:10.1103/physrevlett.81.1662 1998

  24. [24]

    Onida , author L

    author author G. Onida , author L. Reining , author R. W. \ Godby , author R. Del Sole ,\ and\ author W. Andreoni ,\ title title Ab initio calculations of the quasiparticle and absorption spectra of clusters: The sodium tetramer , \ https://doi.org/10.1103/PhysRevLett.75.818 journal journal Phys. Rev. Lett. \ volume 75 ,\ pages 818--821 ( year 1995 ) NoStop

    work page doi:10.1103/physrevlett.75.818 1995

  25. [25]

    Ishii , author K

    author author S. Ishii , author K. Ohno , author Y. Kawazoe ,\ and\ author S. G. \ Louie ,\ title title Ab initio GW quasiparticle energies of small sodium clusters by an all-electron mixed-basis approach , \ https://doi.org/10.1103/PhysRevB.63.155104 journal journal Phys. Rev. B \ volume 63 ,\ pages 155104 ( year 2001 ) NoStop

    work page doi:10.1103/physrevb.63.155104 2001

  26. [26]

    Ishii , author K

    author author S. Ishii , author K. Ohno , author Y. Kawazoe ,\ and\ author S. G. \ Louie ,\ title title Ab initio GW quasiparticle calculation of small alkali-metal clusters , \ https://doi.org/10.1103/PhysRevB.65.245109 journal journal Phys. Rev. B \ volume 65 ,\ pages 245109 ( year 2002 ) NoStop

    work page doi:10.1103/physrevb.65.245109 2002

  27. [27]

    Rohlfing , author P

    author author M. Rohlfing , author P. Kr \"u ger ,\ and\ author J. Pollmann ,\ title title Efficient scheme for GW quasiparticle band-structure calculations with applications to bulk Si and to the Si (001)-(2 1) surface , \ https://doi.org/10.1103/PhysRevB.52.1905 journal journal Phys. Rev. B \ volume 52 ,\ pages 1905--1917 ( year 1995 ) NoStop

    work page doi:10.1103/physrevb.52.1905 1905

  28. [28]

    author author J. C. \ Grossman , author M. Rohlfing , author L. Mitas , author S. G. \ Louie ,\ and\ author M. L. \ Cohen ,\ title title High accuracy many-body calculational approaches for excitations in molecules , \ https://doi.org/10.1103/PhysRevLett.86.472 journal journal Phys. Rev. Lett. \ volume 86 ,\ pages 472--475 ( year 2001 ) NoStop

    work page doi:10.1103/physrevlett.86.472 2001

  29. [29]

    author author M. L. \ Tiago , author J. E. \ Northrup ,\ and\ author S. G. \ Louie ,\ title title Ab initio calculation of the electronic and optical properties of solid pentacene , \ https://doi.org/10.1103/PhysRevB.67.115212 journal journal Phys. Rev. B \ volume 67 ,\ pages 115212 ( year 2003 ) NoStop

    work page doi:10.1103/physrevb.67.115212 2003

  30. [30]

    author author M. L. \ Tiago \ and\ author J. R. \ Chelikowsky ,\ title title Optical excitations in organic molecules, clusters, and defects studied by first-principles Green 's function methods , \ https://doi.org/10.1103/PhysRevB.73.205334 journal journal Phys. Rev. B \ volume 73 ,\ pages 205334 ( year 2006 ) NoStop

    work page doi:10.1103/physrevb.73.205334 2006

  31. [31]

    author author M. L. \ Tiago , author P. R. C. \ Kent , author R. Q. \ Hood ,\ and\ author F. A. \ Reboredo ,\ title title Neutral and charged excitations in carbon fullerenes from first-principles many-body theories , \ https://doi.org/10.1063/1.2973627 journal journal J. Chem. Phys. \ volume 129 ,\ pages 084311 ( year 2008 ) NoStop

    work page doi:10.1063/1.2973627 2008

  32. [32]

    Rostgaard, K

    author author C. Rostgaard , author K. W. \ Jacobsen ,\ and\ author K. S. \ Thygesen ,\ title title Fully self-consistent GW calculations for molecules , \ https://doi.org/10.1103/PhysRevB.81.085103 journal journal Phys. Rev. B \ volume 81 ,\ pages 085103 ( year 2010 ) NoStop

    work page doi:10.1103/physrevb.81.085103 2010

  33. [33]

    Blase , author C

    author author X. Blase , author C. Attaccalite ,\ and\ author V. Olevano ,\ title title First-principles GW calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications , \ https://doi.org/10.1103/PhysRevB.83.115103 journal journal Phys. Rev. B \ volume 83 ,\ pages 115103 ( year 2011 ) NoStop

    work page doi:10.1103/physrevb.83.115103 2011

  34. [34]

    Faber , author C

    author author C. Faber , author C. Attaccalite , author V. Olevano , author E. Runge ,\ and\ author X. Blase ,\ title title First-principles GW calculations for DNA and RNA nucleobases , \ https://doi.org/10.1103/PhysRevB.83.115123 journal journal Phys. Rev. B \ volume 83 ,\ pages 115123 ( year 2011 ) NoStop

    work page doi:10.1103/physrevb.83.115123 2011

  35. [35]

    author author S.-H. \ Ke ,\ title title All-electron GW methods implemented in molecular orbital space: Ionization energy and electron affinity of conjugated molecules , \ https://doi.org/10.1103/PhysRevB.84.205415 journal journal Phys. Rev. B \ volume 84 ,\ pages 205415 ( year 2011 ) NoStop

    work page doi:10.1103/physrevb.84.205415 2011

  36. [36]

    author author F. Bruneval ,\ title title Ionization energy of atoms obtained from GW self-energy or from random phase approximation total energies , \ https://doi.org/10.1063/1.4718428 journal journal J. Chem. Phys. \ volume 136 ,\ pages 194107 ( year 2012 ) NoStop

    work page doi:10.1063/1.4718428 2012

  37. [37]

    K \"o rzd \"o rfer \ and\ author N

    author author T. K \"o rzd \"o rfer \ and\ author N. Marom ,\ title title Strategy for finding a reliable starting point for G_0W_0 demonstrated for molecules , \ https://doi.org/10.1103/PhysRevB.86.041110 journal journal Phys. Rev. B \ volume 86 ,\ pages 041110 ( year 2012 ) NoStop

    work page doi:10.1103/physrevb.86.041110 2012

  38. [38]

    Umari , author L

    author author P. Umari , author L. Giacomazzi , author F. De Angelis , author M. Pastore ,\ and\ author S. Baroni ,\ title title Energy-level alignment in organic dye-sensitized TiO2 from GW calculations , \ https://doi.org/10.1063/1.4809994 journal journal J. Chem. Phys. \ volume 139 ,\ pages 014709 ( year 2013 ) NoStop

    work page doi:10.1063/1.4809994 2013

  39. [39]

    Brooks , author G

    author author J. Brooks , author G. Weng , author S. Taylor ,\ and\ author V. Vlcek ,\ title title Stochastic many-body perturbation theory for moir \'e states in twisted bilayer phosphorene , \ https://doi.org/10.1088/1361-648X/ab6d8c journal journal J. Phys. Condens. Matter \ volume 32 ,\ pages 234001 ( year 2020 ) NoStop

    work page doi:10.1088/1361-648x/ab6d8c 2020

  40. [40]

    Graml , author K

    author author M. Graml , author K. Zollner , author D. Hernang \'o mez-P \'e rez , author P. E. \ Faria Junior ,\ and\ author J. Wilhelm ,\ title title Low-scaling GW algorithm applied to twisted transition-metal dichalcogenide heterobilayers , \ https://doi.org/10.1021/acs.jctc.3c01230 journal journal J. Chem. Theory Comput. \ volume 20 ,\ pages 2202--22...

    work page doi:10.1021/acs.jctc.3c01230 2024

  41. [41]

    F \"o rster \ and\ author F

    author author A. F \"o rster \ and\ author F. Bruneval ,\ title title Why does the GW approximation give accurate quasiparticle energies? The cancellation of vertex corrections quantified , \ https://doi.org/10.1021/acs.jpclett.4c03126 journal journal J. Phys. Chem. Lett. \ volume 15 ,\ pages 12526--12534 ( year 2024 ) NoStop

    work page doi:10.1021/acs.jpclett.4c03126 2024

  42. [42]

    author author E. L. \ Shirley ,\ title title Self-consistent GW and higher-order calculations of electron states in metals , \ https://doi.org/10.1103/PhysRevB.54.7758 journal journal Phys. Rev. B \ volume 54 ,\ pages 7758--7764 ( year 1996 ) NoStop

    work page doi:10.1103/physrevb.54.7758 1996

  43. [43]

    Del Sole , author L

    author author R. Del Sole , author L. Reining ,\ and\ author R. W. \ Godby ,\ title title GW approximation for electron self-energies in semiconductors and insulators , \ https://doi.org/10.1103/PhysRevB.49.8024 journal journal Phys. Rev. B \ volume 49 ,\ pages 8024--8028 ( year 1994 ) NoStop

    work page doi:10.1103/physrevb.49.8024 1994

  44. [44]

    Schindlmayr \ and\ author R

    author author A. Schindlmayr \ and\ author R. W. \ Godby ,\ title title Systematic vertex corrections through iterative solution of Hedin 's equations beyond the GW approximation , \ https://doi.org/10.1103/PhysRevLett.80.1702 journal journal Phys. Rev. Lett. \ volume 80 ,\ pages 1702--1705 ( year 1998 ) NoStop

    work page doi:10.1103/physrevlett.80.1702 1998

  45. [45]

    author author A. J. \ Morris , author M. Stankovski , author K. T. \ Delaney , author P. Rinke , author P. Garc\' a-Gonz\'alez ,\ and\ author R. W. \ Godby ,\ title title Vertex corrections in localized and extended systems , \ https://doi.org/10.1103/PhysRevB.76.155106 journal journal Phys. Rev. B \ volume 76 ,\ pages 155106 ( year 2007 ) NoStop

    work page doi:10.1103/physrevb.76.155106 2007

  46. [46]

    Shishkin, M

    author author M. Shishkin , author M. Marsman ,\ and\ author G. Kresse ,\ title title Accurate quasiparticle spectra from self-consistent GW calculations with vertex corrections , \ https://doi.org/10.1103/PhysRevLett.99.246403 journal journal Phys. Rev. Lett. \ volume 99 ,\ pages 246403 ( year 2007 ) NoStop

    work page doi:10.1103/physrevlett.99.246403 2007

  47. [47]

    Romaniello , author S

    author author P. Romaniello , author S. Guyot ,\ and\ author L. Reining ,\ title title The self-energy beyond GW : Local and nonlocal vertex corrections , \ https://doi.org/10.1063/1.3249965 journal journal J. Chem. Phys. \ volume 131 ,\ pages 154111 ( year 2009 ) NoStop

    work page doi:10.1063/1.3249965 2009

  48. [48]

    Romaniello , author F

    author author P. Romaniello , author F. Bechstedt ,\ and\ author L. Reining ,\ title title Beyond the GW approximation: Combining correlation channels , \ https://doi.org/10.1103/PhysRevB.85.155131 journal journal Phys. Rev. B \ volume 85 ,\ pages 155131 ( year 2012 ) NoStop

    work page doi:10.1103/physrevb.85.155131 2012

  49. [49]

    Gr\"uneis , author G

    author author A. Gr\"uneis , author G. Kresse , author Y. Hinuma ,\ and\ author F. Oba ,\ title title Ionization potentials of solids: The importance of vertex corrections , \ https://doi.org/10.1103/PhysRevLett.112.096401 journal journal Phys. Rev. Lett. \ volume 112 ,\ pages 096401 ( year 2014 ) NoStop

    work page doi:10.1103/physrevlett.112.096401 2014

  50. [50]

    Hung , author F

    author author L. Hung , author F. Bruneval , author K. Baishya ,\ and\ author S. \"O g \"u t ,\ title title Benchmarking the GW approximation and Bethe--Salpeter Equation for groups IB and IIB atoms and monoxides , \ https://doi.org/10.1021/acs.jctc.7b00123 journal journal J. Chem. Theory Comput. \ volume 13 ,\ pages 2135--2146 ( year 2017 ) NoStop

    work page doi:10.1021/acs.jctc.7b00123 2017

  51. [51]

    Maggio \ and\ author G

    author author E. Maggio \ and\ author G. Kresse ,\ title title GW vertex corrected calculations for molecular systems , \ https://doi.org/10.1021/acs.jctc.7b00586 journal journal J. Chem. Theory Comput. \ volume 13 ,\ pages 4765--4778 ( year 2017 ) NoStop

    work page doi:10.1021/acs.jctc.7b00586 2017

  52. [52]

    Wang , author P

    author author Y. Wang , author P. Rinke ,\ and\ author X. Ren ,\ title title Assessing the G_0W_0 _0 (1) Approach : Beyond G_0W_0 with Hedin 's Full Second-Order Self-Energy Contribution , \ https://doi.org/10.1021/acs.jctc.1c00488 journal journal J. Chem. Theory Comput. \ volume 17 ,\ pages 5140--5154 ( year 2021 ) NoStop

    work page doi:10.1021/acs.jctc.1c00488 2021

  53. [53]

    Mejuto-Zaera \ and\ author V

    author author C. Mejuto-Zaera \ and\ author V. Vl c ek ,\ title title Self-consistency in GW formalism leading to quasiparticle-quasiparticle couplings , \ https://doi.org/10.1103/PhysRevB.106.165129 journal journal Phys. Rev. B \ volume 106 ,\ pages 165129 ( year 2022 ) NoStop

    work page doi:10.1103/physrevb.106.165129 2022

  54. [54]

    F \"o rster \ and\ author L

    author author A. F \"o rster \ and\ author L. Visscher ,\ title title Exploring the statically screened G_3W_2 correction to the GW self-energy: Charged excitations and total energies of finite systems , \ https://doi.org/10.1103/PhysRevB.105.125121 journal journal Phys. Rev. B \ volume 105 ,\ pages 125121 ( year 2022 b ) NoStop

    work page doi:10.1103/physrevb.105.125121 2022

  55. [55]

    Weng , author R

    author author G. Weng , author R. Mallarapu ,\ and\ author V. Vl c ek ,\ title title Embedding vertex corrections in GW self-energy: Theory , implementation, and outlook , \ https://doi.org/10.1063/5.0139117 journal journal . Chem. Phys. \ volume 158 ,\ pages 144105 ( year 2023 ) NoStop

    work page doi:10.1063/5.0139117 2023

  56. [56]

    Wen , author V

    author author M. Wen , author V. Abraham , author G. Harsha , author A. Shee , author K. B. \ Whaley ,\ and\ author D. Zgid ,\ title title Comparing self-consistent GW and vertex-corrected G_0W_0 ( G_0W_0 ) accuracy for molecular ionization potentials , \ https://doi.org/10.1021/acs.jctc.3c01279 journal journal J. Chem. Theo. Comput. \ volume 20 ,\ pages ...

    work page doi:10.1021/acs.jctc.3c01279 2024

  57. [57]

    Bruneval \ and\ author A

    author author F. Bruneval \ and\ author A. F \"o rster ,\ title title Fully dynamic G_3W_2 self-energy for finite systems: Formulas and benchmark , \ https://doi.org/10.1021/acs.jctc.4c00090 journal journal J. Chem. Theory Comput. \ volume 20 ,\ pages 3218--3230 ( year 2024 ) NoStop

    work page doi:10.1021/acs.jctc.4c00090 2024

  58. [58]

    F \"o rster ,\ title title Beyond Quasi-Particle Self-Consistent GW for Molecules with Vertex Corrections , \ https://doi.org/10.1021/acs.jctc.4c01639 journal journal J

    author author A. F \"o rster ,\ title title Beyond quasi-particle self-consistent GW for molecules with vertex corrections , \ https://doi.org/10.1021/acs.jctc.4c01639 journal journal J. Chem. Theory Comput. \ volume 21 ,\ pages 1709--1721 ( year 2025 ) NoStop

    work page doi:10.1021/acs.jctc.4c01639 2025

  59. [59]

    author author A. M. \ Lewis \ and\ author T. C. \ Berkelbach ,\ title title Vertex corrections to the polarizability do not improve the GW approximation for the ionization potential of molecules , \ https://doi.org/10.1021/acs.jctc.8b00995 journal journal J. Chem. Theory Comput. \ volume 15 ,\ pages 2925 ( year 2019 ) NoStop

    work page doi:10.1021/acs.jctc.8b00995 2019

  60. [60]

    Verdozzi , author R

    author author C. Verdozzi , author R. W. \ Godby ,\ and\ author S. Holloway ,\ title title Evaluation of GW approximations for the self-energy of a Hubbard cluster , \ https://doi.org/10.1103/PhysRevLett.74.2327 journal journal Phys. Rev. Lett. \ volume 74 ,\ pages 2327--2330 ( year 1995 ) NoStop

    work page doi:10.1103/physrevlett.74.2327 1995

  61. [61]

    Di Sabatino , author J

    author author S. Di Sabatino , author J. A. \ Berger , author L. Reining ,\ and\ author P. Romaniello ,\ title title Photoemission spectra from reduced density matrices: The band gap in strongly correlated systems , \ https://doi.org/10.1103/PhysRevB.94.155141 journal journal Physical Review B \ volume 94 ,\ pages 155141 ( year 2016 ) NoStop

    work page doi:10.1103/physrevb.94.155141 2016

  62. [62]

    author author J. M. \ Tomczak , author P. Liu , author A. Toschi , author G. Kresse ,\ and\ author K. Held ,\ title title Merging GW with DMFT and non-local correlations beyond , \ https://doi.org/10.1140/epjst/e2017-70053-1 journal journal Eur. Phys. J. Spec. Top. \ volume 226 ,\ pages 2565--2590 ( year 2017 ) NoStop

    work page doi:10.1140/epjst/e2017-70053-1 2017

  63. [63]

    Dvorak \ and\ author P

    author author M. Dvorak \ and\ author P. Rinke ,\ title title Dynamical configuration interaction: Quantum embedding that combines wave functions and Green's functions , \ https://doi.org/10.1103/PhysRevB.99.115134 journal journal Phys. Rev. B \ volume 99 ,\ pages 115134 ( year 2019 ) NoStop

    work page doi:10.1103/physrevb.99.115134 2019

  64. [64]

    Quantum embedding theory in the screened Coulomb interaction: Combining configuration interaction withGW/BSE.Phys

    author author M. Dvorak , author D. Golze ,\ and\ author P. Rinke ,\ title title Quantum embedding theory in the screened Coulomb interaction: Combining configuration interaction with GW / BSE , \ https://doi.org/10.1103/PhysRevMaterials.3.070801 journal journal Phys. Rev. Mat. \ volume 3 ,\ pages 070801(R) ( year 2019 ) NoStop

    work page doi:10.1103/physrevmaterials.3.070801 2019

  65. [65]

    Di Sabatino , author J

    author author S. Di Sabatino , author J. Koskelo , author J. A. \ Berger ,\ and\ author P. Romaniello ,\ title title Introducing screening in one-body density matrix functionals: Impact on charged excitations of model systems via the extended Koopmans' theorem , \ https://doi.org/10.1103/PhysRevB.105.235123 journal journal Phys. Rev. B \ volume 105 ,\ pag...

    work page doi:10.1103/physrevb.105.235123 2022

  66. [66]

    Di Sabatino , author J

    author author S. Di Sabatino , author J. Koskelo , author J. A. \ Berger ,\ and\ author P. Romaniello ,\ title title Screened extended Koopmans' theorem: Photoemission at weak and strong correlation , \ https://doi.org/10.1103/PhysRevB.107.035111 journal journal Phys. Rev. B \ volume 107 ,\ pages 035111 ( year 2023 ) NoStop

    work page doi:10.1103/physrevb.107.035111 2023

  67. [67]

    Orlando , author P

    author author R. Orlando , author P. Romaniello ,\ and\ author P.-F. \ Loos ,\ title title The three channels of many-body perturbation theory: GW , particle-particle, and electron-hole T -matrix self-energies , \ https://doi.org/10.1063/5.0176898 journal journal J. Chem. Phys. \ volume 159 ,\ pages 184113 ( year 2023 ) NoStop

    work page doi:10.1063/5.0176898 2023

  68. [68]

    Ammar , author A

    author author A. Ammar , author A. Marie , author M. Rodr \' guez-Mayorga , author H. G. A. \ Burton ,\ and\ author P.-F. \ Loos ,\ title title Can GW handle multireference systems? \ https://doi.org/10.1063/5.0196561 journal journal J. Chem. Phys. \ volume 160 ,\ pages 114101 ( year 2024 ) NoStop

    work page doi:10.1063/5.0196561 2024

  69. [69]

    De Dominicis \ and\ author P

    author author C. De Dominicis \ and\ author P. C. \ Martin ,\ title title Stationary entropy principle and renormalization in normal and superfluid systems. I. Algebraic formulation , \ https://doi.org/10.1063/1.1704062 journal journal J. Math. Phys. \ volume 5 ,\ pages 14--30 ( year 1964 a ) NoStop

    work page doi:10.1063/1.1704062 1964

  70. [70]

    De Dominicis \ and\ author P

    author author C. De Dominicis \ and\ author P. C. \ Martin ,\ title title Stationary entropy principle and renormalization in normal and superfluid systems. II. Diagrammatic formulation , \ https://doi.org/10.1063/1.1704064 journal journal J. Math. Phys. \ volume 5 ,\ pages 31--59 ( year 1964 b ) NoStop

    work page doi:10.1063/1.1704064 1964

  71. [71]

    Baym \ and\ author L

    author author G. Baym \ and\ author L. P. \ Kadanoff ,\ title title Conservation laws and correlation functions , \ https://doi.org/10.1103/PhysRev.124.287 journal journal Phys. Rev. \ volume 124 ,\ pages 287--299 ( year 1961 ) NoStop

    work page doi:10.1103/physrev.124.287 1961

  72. [72]

    Danielewicz ,\ title title Quantum theory of nonequilibrium processes, I , \ https://doi.org/https://doi.org/10.1016/0003-4916(84)90092-7 journal journal Ann

    author author P. Danielewicz ,\ title title Quantum theory of nonequilibrium processes, I , \ https://doi.org/https://doi.org/10.1016/0003-4916(84)90092-7 journal journal Ann. Phys. \ volume 152 ,\ pages 239--304 ( year 1984 ) NoStop

    work page doi:10.1016/0003-4916(84)90092-7 1984

  73. [73]

    author author H. A. \ Bethe \ and\ author J. Goldstone ,\ title title Effect of a repulsive core in the theory of complex nuclei , \ http://www.jstor.org/stable/100108 journal journal Proc. Math. Phys. Eng. Sci. \ volume 238 ,\ pages 551--567 ( year 1957 ) NoStop

    work page 1957

  74. [74]

    author author A. L. \ Fetter \ and\ author J. D. \ Walecka ,\ @noop title Quantum Theory of Many Particle Systems \ ( publisher McGraw Hill, San Francisco ,\ year 1971 ) NoStop

    work page 1971

  75. [75]

    Marie , author P

    author author A. Marie , author P. Romaniello ,\ and\ author P.-F. \ Loos ,\ title title Anomalous propagators and the particle-particle channel: Hedin 's equations , \ https://doi.org/10.1103/PhysRevB.110.115155 journal journal Phys. Rev. B \ volume 110 ,\ pages 115155 ( year 2024 ) NoStop

    work page doi:10.1103/physrevb.110.115155 2024

  76. [76]

    Zhang , author N

    author author D. Zhang , author N. Q. \ Su ,\ and\ author W. Yang ,\ title title Accurate quasiparticle spectra from the T -matrix self-energy and the particle-particle random phase approximation , \ https://doi.org/10.1021/acs.jpclett.7b01275 journal journal J. Phys. Chem. Lett. \ volume 8 ,\ pages 3223--3227 ( year 2017 ) NoStop

    work page doi:10.1021/acs.jpclett.7b01275 2017

  77. [77]

    Monino \ and\ author P.-F

    author author E. Monino \ and\ author P.-F. \ Loos ,\ title title Connections and performances of Green's function methods for charged and neutral excitations , \ https://doi.org/10.1063/5.0159853 journal journal J. Chem. Phys. \ volume 159 ,\ pages 034105 ( year 2023 ) NoStop

    work page doi:10.1063/5.0159853 2023

  78. [78]

    Marie \ and\ author P.-F

    author author A. Marie \ and\ author P.-F. \ Loos ,\ title title Reference energies for valence ionizations and satellite transitions , \ https://doi.org/10.1021/acs.jctc.4c00216 journal journal J. Chem. Theory Comput. \ volume 20 ,\ pages 4751--4777 ( year 2024 ) NoStop

    work page doi:10.1021/acs.jctc.4c00216 2024

  79. [79]

    Liebsch ,\ title title Ni d -band self-energy beyond the low-density limit , \ https://doi.org/10.1103/PhysRevB.23.5203 journal journal Phys

    author author A. Liebsch ,\ title title Ni d -band self-energy beyond the low-density limit , \ https://doi.org/10.1103/PhysRevB.23.5203 journal journal Phys. Rev. B \ volume 23 ,\ pages 5203--5212 ( year 1981 ) NoStop

    work page doi:10.1103/physrevb.23.5203 1981

  80. [80]

    Springer , author F

    author author M. Springer , author F. Aryasetiawan ,\ and\ author K. Karlsson ,\ title title First-principles T -matrix theory with application to the 6 eV satellite in Ni , \ https://doi.org/10.1103/PhysRevLett.80.2389 journal journal Phys. Rev. Lett. \ volume 80 ,\ pages 2389--2392 ( year 1998 ) NoStop

    work page doi:10.1103/physrevlett.80.2389 1998

Showing first 80 references.