Approaching Coupled Cluster Accuracy with Positive Semidefinite Vertex Corrected Self-Energies

Arno F\"{o}rster; Fabien Bruneval; Yaroslav Pavlyukh

REVIEW 3 major objections 4 minor 65 references

Approaching Coupled Cluster Accuracy with Positive Semidefinite Vertex Corrected Self-Energies

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Completing the square of a diagrammatic self-energy restores spectral positivity and reaches near coupled-cluster accuracy for ionization energies.

desk verdict Clean formal fix for the BSE self-energy's negative spectral weight, with real ~0.1 eV HOMO gains on a selected reference — but the 'consistently improves' framing overreaches once you look at HOMO-1 or other starting points. read the letter →

arxiv 2607.29359 v1 pith:HTSJTBBG submitted 2026-07-31 physics.chem-ph

Yaroslav Pavlyukh , Fabien Bruneval , Arno Förster This is my paper · ORCID

classification physics.chem-ph

keywords Bethe-Salpeterequationself-energypositivesemidefinitespectralfunctionGWapproximationnonequilibriumGreen'sfunctionsquasiparticleionizationenergiescoupled-clusterbenchmark

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that the negative spectral weight produced by a Bethe-Salpeter-equation-based self-energy can be eliminated by a positive semidefinite (PSD) completion: recombining the diagram's two half-diagrams into a complete square. The minimal completion, Σ_PSD-I, doubles the exchange term and adds a particle-hole T-matrix self-energy built from singlet BSE excitations, turning negative dips into real satellite peaks. Benchmarked on 23 small molecules and on the 100-molecule GW100 set, the corrected self-energy gives HOMO ionization energies with a mean absolute deviation of about 0.1 eV from near-full-CI references, close to the intrinsic accuracy of EOM-CCSD, at essentially the cost of the parent BSE-based scheme. The point is that a formal positivity constraint doubles as an accuracy improvement, not just a cure for an artifact.

What carries the argument

The load-bearing device is the PSD completion of half-diagrams on the Keldysh contour. Cutting the exchange diagram at the boundary between the minus and plus branches gives two half-diagrams a^ν_kσ and b^ν_kσ (Eqs. 4) expressed through BSE transition amplitudes; a self-energy built as the complete square (a+b)(a+b) = a·a + 2a·b + b·b has rates in Fermi golden rule form, hence a nonnegative spectral function. The new b·b term is a particle-hole T-matrix self-energy Σ_Tph(S) from singlet BSE excitation energies Ω_ν and amplitudes Z^ν, while the doubled a·b term restores the original exchange contribution. Because the statically screened interaction W0 is kept distinct from the bare Coulomb v,

What would settle it

Find a molecule where the static-kernel BSE singlet excitation energies carry a bias comparable to the known triplet bias (for example, a strong charge-transfer or double-excitation case). If Σ_PSD-I's HOMO MAD there degrades into the 0.2–0.5 eV range typical of Σ_PSD-II, the claimed accuracy would be shown to be inherited from the input manifold rather than produced by the PSD construction. A direct spectral-function calculation that exhibits residual negative weight on any molecule would also falsify the positivity guarantee.

Watch

Extended reading notes

Core claim

The authors derive the minimal positive semidefinite extension of the BSE-based self-energy. Using the nonequilibrium Green's function formalism, they cut the exchange-type self-energy diagram along its Keldysh-contour boundary into two half-diagrams a and b, built from Coulomb and screened-Coulomb vertices with BSE transition amplitudes, and recombine them as the complete square (a+b)(a+b). This yields Σ_PSD-I = Σ_d + 2Σ_2x + Σ_Tph(S), where the new particle-hole T-matrix term Σ_Tph(S) uses singlet BSE excited states. The completed square guarantees a nonnegative rate function and a nonnegative spectral function, converts the parent theory's small negative spectral weight into positive sate

Load-bearing premise

The strongest result depends on two premises: the static-kernel BSE singlet excitation manifold is accurate enough for the recombined self-energy to inherit its quality, and the non-self-consistent HF+γGW reference protocol is a valid starting point—on plain HF, Σ_PSD-I does not improve over the parent theory.

Editorial extensions

If this is right

The BSE-based self-energy can be made spectral-function-positive without sacrificing efficiency: Σ_PSD-I costs essentially the same as the parent BSE-based self-energy.
On the 23-molecule set, Σ_PSD-I with a density-matrix-corrected HF reference gives a HOMO MAD of 0.095 eV, versus 0.082 eV for EOM-CCSD and 0.43 eV for one-shot GW.
On the GW100 set, Σ_PSD-I yields a HOMO MAD of 0.10 eV, roughly halving the GW error and improving on the parent Σ_BSE (0.15 eV).
Admitting triplet intermediate states (Σ_PSD-II) degrades results (MADs 0.2–0.5 eV) because the static-kernel BSE underestimates triplet excitation energies; the singlet-only minimal completion is therefore the recommended scheme.
The parent theory's negative spectral weight reappears as small positive satellite peaks at nearly the same energies, enriching the satellite spectrum.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Since the PSD completion only rearranges existing diagrams, its accuracy is bounded by the quality of the input singlet BSE excitation manifold; improving the triplet kernel (e.g., with dynamical screening) should directly unlock Σ_PSD-II's potential.
The same half-diagram square-completion logic could apply to other vertex corrections or cumulant-type satellite constructions, where positivity of the spectral function is also a requirement.
The strong reference sensitivity (on plain HF, Σ_PSD-I does not beat the parent scheme) indicates that starting-point optimization remains a controlling factor; a self-consistent evaluation could remove this dependence.
The demonstrated accuracy on light molecules invites testing on extended systems and on spectra beyond valence ionization (electron affinities, core levels), where positive-definiteness of the spectral function is equally mandatory.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The thing to know: this paper supplies a genuinely clean fix for a real flaw in the BSE-based self-energy of Ref. 1 — the negative spectral weight — and shows that on one protocol (HF+γGW) the fixed self-energy gets HOMO MADs near 0.1 eV on both Marie-Loos and GW100, close to EOM-CCSD. The other thing to know: the abstract's "consistently improves quasiparticle energies" is not supported by the paper's own numbers. The improvement is real for HOMO at the recommended reference, and much weaker elsewhere.

The genuinely new content is the derivation of the PSD completions Σ_PSD-I and Σ_PSD-II, plus the working equations (13) and (15). The PSD framework is prior work (Stefanucci and Pavlyukh), but applying it to this self-energy is new, and the construction is a formal identity — a complete square, Eq. (7) — with no fitted constants. The spin analysis is rigorous: only singlets enter Σ_d and Σ_2x, so the minimal completion Σ_PSD-I is the natural choice, and the triplet channel degrades results exactly as expected given the static-kernel BSE's known triplet bias. Fig. 5 shows the negative dips in the rate function becoming small positive satellites. The GW100 test is a genuine out-of-sample check for HOMO, and the error distribution is narrow and centered at zero.

The soft spots, in order. First, property selectivity: on HOMO-1 under the recommended HF+γGW protocol, PSD-I gives 0.288 eV MAD against 0.239 for the parent Σ_BSE and 0.164 for EOM-CCSD. The text admits the HOMO-1 advantage is "less systematic," but the abstract and discussion generalize anyway. The construction also inherits the input manifold's errors; the triplet failure shows how sensitive it is. Second, reference selection: HF+γGW was picked after seeing Marie-Loos performance, and GW100 only validates HOMO. Legitimate, though not fatal — the GW100 confirmation is real. Third, on plain HF, PSD-I (0.307 eV) doesn't beat Σ_BSE (0.280 eV), and on RSH+γGW, plain GW (0.105 eV) beats PSD-I (0.134 eV); the RSH comparison is muddied by per-method optimal tuning. None of this sinks the recommended protocol, but it does sink the broad framing. Fourth, no uncertainty estimates, so the sub-0.02 eV differences among PSD variants are not meaningful. Fifth, minor: the Fig. 6/7 captions call the Marie-Loos reference CCSD(T)ΔSCF when it is CIPSI, say "23 molecules" when two are omitted, and the code is only "available upon request."

Who gets value: anyone working on vertex corrections, spectral functions, or molecular GW/BSE. It deserves a serious referee — the math holds up and the HOMO result is interesting. I'd send it out, asking for a scoped abstract, confidence intervals, corrected captions, and the code.

Referee Report

3 major / 4 minor

Summary. The paper derives positive-semidefinite (PSD) extensions of the BSE-based self-energy of Ref. 1. Using the NEGF partitioning of Ref. 31, the exchange diagram is split into half-diagrams a and b, and the minimal complete square (a+b)(a+b) yields Eq. (7). Restricting the intermediate excitations to singlets gives PSD-I (Eq. 11), and including triplets gives PSD-II (Eq. 14). The authors implement these self-energies in MOLGW and benchmark them on the Marie–Loos set (23 molecules) and GW100 (100 molecules) for HOMO and HOMO-1 ionization energies. They report that PSD-I on an HF+γGW reference achieves HOMO MADs of 0.095 eV and 0.10 eV on the two sets, close to EOM-CCSD, while restoring spectral-function positivity. The formal derivation is internally consistent, but the empirical claims are narrower than the abstract and discussion suggest: the improvement is HOMO-specific, reference-dependent, and the HOMO-1 results under the recommended protocol are worse than the parent BSE self-energy.

Significance. If the claims were fully validated, this would be a valuable contribution: it provides a principled, parameter-free completion of the BSE-based self-energy that guarantees a nonnegative spectral function and adds a physical T-matrix channel, rather than a damped ad hoc correction. The formal construction (Eqs. 7, 11) is elegant and machine-checkable in the sense of following from explicit half-diagram algebra, and the GW100 out-of-sample HOMO test is a useful sanity check. However, the significance of the numerical claim is currently limited by reference selection and by the absence of evidence for deeper valence states. The paper's central conclusion—'consistently improves quasiparticle energies' and 'accuracy comparable to coupled-cluster'—is supported only for HOMO at one performance-selected reference, which weakens the claim from a general methodological statement to a benchmark-specific observation.

major comments (3)

[Abstract; Discussion; Figs. 6–7] The Abstract and Discussion claim 'consistently improves quasiparticle energies' and 'accuracy comparable to coupled-cluster reference calculations.' But under the recommended HF+γGW protocol, Fig. 7 gives PSD-I HOMO-1 MAD = 0.288 eV, which is worse than the parent Σ_BSE (0.239 eV) and far from EOM-CCSD (0.164 eV). The Discussion sentence correctly says 'HOMO energies,' yet the broader wording in the Abstract is not supported for valence ionization energies generally. Please restrict the claims to HOMO or provide evidence that the improvement carries over to deeper valence states.
[Numerical Results; Fig. 6; 'Synthetic approaches'] The recommended scheme (HF+γGW with PSD-I) was selected after inspecting Marie–Loos performance across four references and several synthetic mixtures. On plain HF, PSD-I (MAD 0.307 eV) does not improve over the parent Σ_BSE (0.280 eV); on RSH+γGW, PSD-I (0.134 eV) is worse than plain GW (0.105 eV). The GW100 out-of-sample test uses only the selected protocol and only HOMO, so it does not validate the selection procedure or the general claim. Unless an a priori criterion for choosing the reference is provided, or a nested training/test split is used, the '0.1 eV accuracy' should be presented as one favorable protocol rather than a robust property of PSD-I.
[Eqs. (3), (11), (14); Numerical Results (PSD-II paragraph)] The PSD construction only rearranges half-diagrams; it cannot repair systematic errors in the input singlet BSE excitation manifold. The triplet-channel failure of PSD-II is correctly attributed to underestimated triplet Ων, but no analogous validation of singlet excitation energies or amplitudes against reference is provided. Since PSD-I inherits all singlet BSE errors, the near-CCSD claim for HOMO is only as good as the static-kernel BSE singlet spectrum. Please either benchmark the singlet intermediates or explicitly state this inherited error as a limitation of the method.

minor comments (4)

[Figs. 6–8 captions] Captions state errors are with respect to 'a reference CCSD(T)ΔSCF evaluation,' while the text says the Marie–Loos reference is CIPSI near-FCI and the GW100 reference is ΔCCSD(T). Please align the captions with the actual reference definitions.
[Abstract] The acronym PSD is used in the Abstract without expansion; spell out 'positive semidefinite' at first use or define it there.
[Computational Details] The basis set is written 'aug-cc-pvQZ'; the standard notation is aug-cc-pVQZ. Also 'LibXClibrary' should be 'libxc.'
[Eq. (14) and text after Eq. (11)] The factor of 3 in the triplet contribution is introduced without explanation. A brief comment that the three degenerate triplet components (Eq. 23) give this multiplicity would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the PSD self-energy is a formal complete-square identity, and the accuracy claims are benchmark outputs, not inputs.

full rationale

The central formal step is the expansion of (a+b)^2: Eq. (7) defines Sigma_PSD as sums of products [a_k^nu + b_k^nu][a_k^nu + b_k^nu], and Eq. (11) is literally that expansion into Sigma_d + 2 Sigma_2x + Sigma_Tph(S). This is an algebraic identity with no fitted constants and no use of the target ionization energies. The paper itself states 'Formal considerations alone, however, cannot single out a preferred member of this family' and deliberately sets the performance question numerically. The numerical claims are tested against independent CIPSI/CCSD(T) references from the Marie-Loos and GW100 benchmarks; the recommended scheme (Sigma_PSD-I on HF+gamma-GW) was selected on Marie-Loos but then evaluated out-of-sample on GW100, so the improvement is not statistically forced by construction. Reliance on prior work by overlapping authors (Refs. 1, 31, 32, 33) is not load-bearing circularity: Ref. 1 is the parent self-energy being improved, Refs. 31/32 are peer-reviewed formal PSD theorems used as algebraic tools, and Ref. 33 is context. The paper also candidly reports the failure modes (triplet PSD-II degradation, HOMO-1 less systematic improvement), which would be pointless if the conclusion were definitional. No circular step can be exhibited.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The PSD completion itself (Eq. 7) is a formal identity with no fitted constants: the construction is exactly (a+b)(a+b). The ledger above captures what the central claim actually rests on: the static BSE kernel (Eq. 3), the strict distinctness of W0 and v (which excludes the TDHF limit and forces hand-chosen admixtures), the spin-restricted closed-shell assumptions, and the asserted no-double-counting property. The empirical freedom is in the model selection - which PSD variant and which reference protocol to recommend - and in the per-method RSH tuning for the RSH columns. No new particles, forces, or conserved quantities are postulated.

free parameters (2)

Synthetic admixture coefficients (1/2, 3/4, and variant-dependent combinations) = 1/2, 3/4, 1/2(Sigma_d + Sigma_Tph), etc. (Table 1)
Chosen by hand so that the PSD family recovers the GW, PT2, and Tph limits in the W0 -> v and W0 -> 0 limits. The paper concedes 'formal considerations alone... cannot single out a preferred member of this family', so the final choice among variants is empirical (Synthetic approaches subsection, Table 1).
Per-method optimal RSH tuning parameters = Not tabulated; determined separately for each self-energy approximation
'The optimal tuning parameters have been determined for each self-energy approximation separately' (Computational Details). Affects only the RSH-based columns of Figs. 6-7; the headline HF+gamma-GW results do not use RSH tuning.

assumptions (5)

domain assumption Static BSE kernel I(1,2,3,4) = delta13 delta24 v(1,4) - delta14 delta23 W0(1,3) (Eq. 3)
Borrowed as 'universally adopted' (Ref. 37); it makes L one-interaction-line reducible and the BSE solvable as a Casida equation. Its known weakness for triplet excitations is documented inside this paper and caps the accuracy of the PSD-II variant.
ad hoc to paper W0 and v are strictly distinct interactions; the W0 -> v (TDHF) limit is excluded
The no-double-counting claim (PSD self-energies subsection) and the Casida-based evaluation both require W0 != v; the TDHF case 'cannot be evaluated within the standard Casida approach' (Ingredients subsection). The excluded limit is then patched by synthetic admixtures.
standard math PSD half-diagram partitioning of Refs. 31-32
The Keldysh-contour partition into islands of pluses/minuses and the complete-square theorem are taken from the authors' earlier published work without re-derivation.
domain assumption Restricted closed-shell spin-adapted reference with real orbitals (Eqs. 22-23)
All formulas assume a spin-compensated singlet ground state; BN and C2 are excluded from the benchmark because they lack a stable singlet mean-field solution. Open-shell or spin-orbit-coupled systems are out of scope.
ad hoc to paper 'As long as W0 is considered distinct from v, there is no double counting at any order'
Asserted in the PSD self-energies subsection without proof. It underpins the physical reading of the new (b,b) T-matrix diagram as new physics rather than an overcount of existing diagrams.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Approaching Coupled Cluster Accuracy with Positive Semidefinite Vertex Corrected Self-Energies." pith.science (2026). https://pith.science/paper/HTSJTBBG

@misc{pith2026260729359,
  author       = {Pith},
  title        = {Pith review of: Approaching Coupled Cluster Accuracy with Positive Semidefinite Vertex Corrected Self-Energies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HTSJTBBG}},
  note         = {Machine review of arXiv:2607.29359}
}

abstract

Hedin's formalism of functional derivatives is the best-known method for systematically constructing correlated electronic theories, largely due to the success of its lowest-order self-energy expansion, the $GW$ approximation. Beyond $GW$, diagrammatic resummation schemes attempt to mix correlations simultaneously across all particle-particle and particle-hole channels. Because such a comprehensive treatment is computationally prohibitive for realistic molecular systems, a highly effective alternative is to fully account for electronic correlations in one specific channel, typically the particle-hole channel. This idea was recently implemented for molecular systems [1], yielding a self-energy expressed in terms of excited-state energies and transition amplitudes from the solution of the Bethe-Salpeter equation, rather than the random phase approximation used in $GW$. While this approach is predictive and numerically efficient, it violates the fundamental positive-definiteness constraint of the electron spectral function in certain energy ranges. In this study, we resolve this physical flaw by deriving a positive semidefinite (PSD) extension of the theory using a rigorous framework based on the nonequilibrium Green's function formalism. The PSD constraint introduces new scattering channels and triplet intermediate states and restores the correct physical behavior. We demonstrate that it consistently improves quasiparticle energies across standard molecular benchmarks, with an accuracy comparable to coupled-cluster reference calculations.

Figures

Figures reproduced from arXiv: 2607.29359 by the authors.

**Figure 1.** Spin resolved self-energy from Ref. 1. Red Greek letters indicate spins. This improvement comes at a cost. Compare the GWA self-energy Σ𝑐 (𝟣, 𝟤) = 𝑖𝐺(𝟣, 𝟤)𝑊 (𝟣, 𝟤) and the BSE-based self-energy Σ𝑐 (𝟣, 𝟤) = 𝑖𝑣(𝟣 + , 𝟥)𝐺(𝟣, 𝟦)𝐼(𝟦, 𝟨, 𝟤, 𝟧)𝐿(𝟧, 𝟥, 𝟨, 𝟥), (1) where 𝗃 ≡ (𝑟𝑗 , 𝜎𝑗 , 𝑡𝑗 ) ≡ (𝑗, 𝜎𝑗 ) and 𝐼(𝟣, 𝟤, 𝟥, 𝟦) = 𝑖𝛿Σ(𝟣, 𝟥)∕𝛿𝐺(𝟦, 𝟤) is the particle-hole irreducible four-point kernel; Eq. (1) is derived in the Methods s… view at source ↗

**Figure 2.** Partition of the two-particle correlation function with degenerate time-arguments 𝑡 2 = 𝑡 4 , 𝑡 1 = 𝑡 3 (left). Irreducible two-particle correlation function (right). Σ < 2x(1, 2) = + 2 − 1 + 4 − 3 𝐿(4,3,2,3) 𝓅4 𝓆𝜈 𝓅𝜈 = 𝑎 ⎧ { { { { { ⎨ { { { { { ⎩ − 1 − 3 𝓆3 𝓅3 𝓅1 𝓅4 𝓅𝜈 𝜈 𝓆𝜈 ⎫ } } } } } ⎬ } } } } } ⎭ × 𝑏 ⎧ { { { { { ⎨ { { { { { ⎩ + 𝓅2 2 𝓆2 𝓆4 + 4 𝓅𝜈 𝓅4 𝜈 𝓆𝜈 ⎫ } } } } } ⎬ } } } } } ⎭ [PITH_FULL_IMAGE:figures/full_fi… view at source ↗

**Figure 3.** Partition of the self-energy diagram resulting from 𝐼(1, 2, 3, 4) = −𝛿(1, 4)𝛿(2, 3)𝑊0 (1, 3). A dashed line denotes the border between the islands of pluses and minuses. 𝐿 is time-degenerate, i.e. 𝑡 2 = 𝑡 4 . External indices are specified as 𝓅1 ≡ (𝑝, 𝛼), 𝓆2 ≡ (𝑞, 𝛼), where 𝑝 and 𝑞 are the orbital indices and 𝛼 is the spin-up index. 𝓅4 = (𝑘, 𝛼) (𝑘 is an occupied state), and 𝛾 label the dangling lines. For internal l… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: PSD self-energy for 𝑊0 distinct from 𝑣 as explicitly given by Eq. (7). 20 18 16 14 12 10 [eV] 0.00 0.02 0.04 0.06 0.08 0.10 ii( ) [e V] BSE PSD I [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: Rate function Γ𝑖𝑖(𝜔) = 𝑖[Σ> 𝑖𝑖(𝜔) − Σ< 𝑖𝑖(𝜔)] for 𝑖 = HOMO of MgO. behavior for 𝑊0 → 0 and 𝑊0 → 𝑣. In order to produce physically meaningful limits we introduce synthetic approaches, which mix several PSD self-energies—all stemming from Eq. (7). Using spin symmetries o…

**Figure 6.** Figure 6: Error distribution of the HOMO energies with respect to a reference CCSD(T) ΔSCF evaluation for the 23 molecules contained in the Marie and Loos set. 9 Different self-energies are evaluated as one-shot perturbations on top of four different reference states. EOM-CCSD (…

**Figure 7.** Figure 7: Error distribution of the HOMO-1 energies with respect to a reference CCSD(T) ΔSCF evaluation for the 23 molecules contained in the Marie and Loos set. 9 Different self-energies are evaluated as one-shot perturbations on top of four different reference states. advantag…

**Figure 8.** Figure 8: Error distribution of the HOMO energies with respect to a reference CCSD(T) ΔSCF evaluation for the 100 molecules contained in the GW100 set. 46 Discussion We have derived positive semidefinite extensions of the BSE-based electron self-energy recently introduced in Ref…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

65 extracted references · 1 linked inside Pith

[1]

Förster, A.; Bruneval, F.J. Phys. Chem. Lett.2024,15, 12526–12534

2024
[2]

Blase, X.; Duchemin, I.; Jacquemin, D.Chemical Society Reviews2018,47, 1022–1043
[3]

Golze, D.; Dvorak, M.; Rinke, P.Frontiers in Chemistry2019,7, 377
[4]

Bruneval, F.; Marques, M. A. L.Journal of Chemical Theory and Computation2013,9, 324–329
[5]

Maggio, E.; Kresse, G.J. Chem. Theory Comput.2017,13, 4765–4778

2017
[6]

Olsen,T.;Patrick,C.E.;Bates,J.E.;Ruzsinszky,A.;Thygesen,K.S.npj Computational Materials2019,5,106
[7]

M.; Berkelbach, T

Lewis, A. M.; Berkelbach, T. C.Journal of Chemical Theory and Computation2019,15, 2925–2932
[8]

J.; Whaley, K

Mejuto-Zaera, C.; Weng, G.; Romanova, M.; Cotton, S. J.; Whaley, K. B.; Tubman, N. M.; Vlček, V.Journal of Chemical Physics2021,154, 121101

Show all 65 references

[9]

Marie, A.; Loos, P.-F.J. Chem. Theory Comput.2024,20, 4751–4777

2024
[10]

G.; Loos, P

Ammar, A.; Marie, A.; Rodríguez-Mayorga, M.; Burton, H. G.; Loos, P. F.Journal of Chemical Physics2024, 160, 114101
[11]

B.; Zgid, D.Journal of Chemical Theory and Compu- tation2024,20, 3109–3120

Wen, M.; Abraham, V.; Harsha, G.; Shee, A.; Whaley, K. B.; Zgid, D.Journal of Chemical Theory and Compu- tation2024,20, 3109–3120
[12]

Abraham, V.; Harsha, G.; Zgid, D.Journal of Chemical Theory and Computation2024,20, 4579–4590
[13]

Marie, A.; Tölle, J.; Loos, P.-F.arXiv:2201.04128 [cond-mat, physics:physics]2026,

2026 arXiv
[14]

J.; Musiał, M.Rev

Bartlett, R. J.; Musiał, M.Rev. Mod. Phys.2007,79, 291–352

2007
[15]

Stuyver, T.; Chen, B.; Zeng, T.; Geerlings, P.; De Proft, F.; Hoffmann, R.Chemical Reviews2019,119, 11291– 11351
[16]

Stefanucci, G.; Leeuwen, R. v.Nonequilibrium many-body theory of quantum systems: a modern introduction, second edition ed.; Cambridge University Press: Cambridge New York Port Melbourne New Delhi Singapore, 2025

2025
[17]

H.; Stoddart, J

Flood, A. H.; Stoddart, J. F.; Steuerman, D. W.; Heath, J. R.Science2004,306, 2055–2056
[18]

A.Science2003,300, 1384–1389

Nitzan, A.; Ratner, M. A.Science2003,300, 1384–1389
[19]

Evers, F.; Korytár, R.; Tewari, S.; Van Ruitenbeek, J. M.Rev. Mod. Phys.2020,92, 035001

2020
[20]

Tuovinen, R.; Pavlyukh, Y.PRX Energy2025,4, 043003
[21]

Rev.1965,139, A796–A823

Hedin, L.Phys. Rev.1965,139, A796–A823

1965
[22]

Rev.1951,84, 1232–1242

Salpeter, E.; Bethe, H.Phys. Rev.1951,84, 1232–1242

1951
[23]

Nuovo Cimento1988,11, 1–86

Strinati, G.Riv. Nuovo Cimento1988,11, 1–86
[24]

Albrecht, S.; Reining, L.; Del Sole, R.; Onida, G.Phys. Rev. Lett.1998,80, 4510

1998
[25]

X.; Shirley, E

Benedict, L. X.; Shirley, E. L.; Bohn, R. B.Phys. Rev. Lett.1998,80, 4514–4517

1998
[26]

Rohlfing, M.; Louie, S. G.Phys. Rev. B2000,62, 4927–4944
[27]

Onida, G.; Reining, L.; Rubio, A.Rev. Mod. Phys.2002,74, 601

2002
[28]

W.; Del Sole, R.; Andreoni, W.Phys

Onida, G.; Reining, L.; Godby, R. W.; Del Sole, R.; Andreoni, W.Phys. Rev. Lett.1995,75, 818. 14

1995
[29]

D.; Reining, L.Rep

Botti, S.; Schindlmayr, A.; Sole, R. D.; Reining, L.Rep. Prog. Phys.2007,70, 357

2007
[30]

Förster, A.; Visscher, L.Journal of Chemical Theory and Computation2022,18, 6779–6793
[31]

Stefanucci, G.; Pavlyukh, Y.; Uimonen, A.-M.; van Leeuwen, R.Phys. Rev. B2014,90, 115134
[32]

Uimonen, A.-M.; Stefanucci, G.; Pavlyukh, Y.; van Leeuwen, R.Phys. Rev. B2015,91, 115104
[33]

Bruneval, F.; Förster, A.; Pavlyukh, Y.J. Chem. Theory Comput.2025, acs.jctc.5c01180

2025
[34]

Pavlyukh, Y.; Uimonen, A.-M.; Stefanucci, G.; van Leeuwen, R.Phys. Rev. Lett.2016,117, 206402

2016
[35]

F.; Marie, A.; Ammar, A.Faraday Discussions2024,254, 240–260

Loos, P. F.; Marie, A.; Ammar, A.Faraday Discussions2024,254, 240–260
[36]

Kockläuner, J.; Golze, D.Journal of Chemical Theory and Computation2025,21, 3101–3119
[37]

M.; Reining, L.; Ceperley, D

Martin, R. M.; Reining, L.; Ceperley, D. M.Interacting electrons; Cambridge University Press, 2016

2016
[38]

Romaniello, P.; Bechstedt, F.; Reining, L.Physical Review B2012,85, 155131
[39]

N.; Ren, X.Physical Review B2019,99, 1–12

Tahir, M. N.; Ren, X.Physical Review B2019,99, 1–12
[40]

Orlando, R.; Romaniello, P.; Loos, P.-F.J. Chem. Phys.2023,159, 184113

2023
[41]

P.; Rancurel, P.The Journal of Chemical Physics1973,5745, 5745–5759

Huron, B.; Malrieu, J. P.; Rancurel, P.The Journal of Chemical Physics1973,5745, 5745–5759
[42]

Refaely-Abramson,S.;Sharifzadeh,S.;Govind,N.;Autschbach,J.;Neaton,J.B.;Baer,R.;Kronik,L.Phys. Rev. Lett.2012,109, 226405

2012
[43]

A.; Hamed, S

McKeon, C. A.; Hamed, S. M.; Bruneval, F.; Neaton, J. B.J. Chem. Phys.2022,157, 074103

2022
[44]

Bruneval, F.Journal of Chemical Theory and Computation2019,15, 4069–4078
[45]

Bruneval, F.Physical Review B2019,99, 041118(R)
[46]

VanSetten,M.J.;Caruso,F.;Sharifzadeh,S.;Ren,X.;Scheffler,M.;Liu,F.;Lischner,J.;Lin,L.;Deslippe,J.R.; Louie,S.G.;Yang,C.;Weigend,F.;Neaton,J.B.;Evers,F.;Rinke,P.J. Chem. Theory Comput.2015,11,5665– 5687

2015
[47]

J.Frontiers in Chemistry2021,9, 749779

Bruneval, F.; Dattani, N.; Van Setten, M. J.Frontiers in Chemistry2021,9, 749779
[48]

Tölle,J.;Kitsaras,M.-P.;Irmler,A.;Grüneis,A.;Loos,P.-F.Journal of Chemical Theory and Computation2026,
[49]

Jacquemin, D.; Duchemin, I.; Blase, X.J. Phys. Chem. Lett.2017,8, 1524–1529

2017
[50]

M.; Bruneval, F.; Neaton, J

Rangel, T.; Hamed, S. M.; Bruneval, F.; Neaton, J. B.Journal of Chemical Physics2017,146, 194108
[51]

S.; Schirmer, J.; Domcke, W.; Von Niessen, W.Journal of Physics B: Atomic and Molecular Physics1977,10, 1–6

Cederbaum, L. S.; Schirmer, J.; Domcke, W.; Von Niessen, W.Journal of Physics B: Atomic and Molecular Physics1977,10, 1–6
[52]

L.Physical Review B2016,94, 155101

Kutepov, A. L.Physical Review B2016,94, 155101
[53]

L.Physical Review B2017,95, 195120

Kutepov, A. L.Physical Review B2017,95, 195120
[54]

Vlček, V.Journal of Chemical Theory and Computation2019,15, 6254–6266
[55]

Vacondio, S.; Varsano, D.; Ruini, A.; Ferretti, A.Journal of Chemical Theory and Computation2024,20, 4718–4737
[56]

H.Journal of Chemical Theory and Computation2024,20, 7479–7493

Patterson, C. H.Journal of Chemical Theory and Computation2024,20, 7479–7493
[57]

Förster, A.Journal of Chemical Theory and Computation2025,21, 1709-1721
[58]

Cunningham,B.;Grüning,M.;Azarhoosh,P.;Pashov,D.;vanSchilfgaarde,M.Physical Review Materials2018, 2, 034603. 15
[59]

Cunningham, B.; Grüning, M.; Pashov, D.; Van Schilfgaarde, M.Physical Review B2023,108, 165104
[60]

Koskelo, J.; Reining, L.; Gatti, M.Physical Review Letters2025,134
[61]

Casida, M. E.J. Mol. Struct. THEOCHEM2009,914, 3–18
[62]

G.; Liu, R.-F.; Toulouse, J.; Jansen, G.J

Ángyán, J. G.; Liu, R.-F.; Toulouse, J.; Jansen, G.J. Chem. Theory Comput.2011,7, 3116–3130

2011
[63]

Baerends, E. J. et al.Journal of Chemical Physics2025,162, 162501
[64]

M.; Shao, M.; Yang, C.; Neaton, J

Bruneval, F.; Rangel, T.; Hamed, S. M.; Shao, M.; Yang, C.; Neaton, J. B.Computer Physics Communications 2016,208, 149–161

2016
[65]

J.; Marques, M.SoftwareX2018,7, 1–5

Lehtola, S.; Steigemann, C.; Oliveira, M. J.; Marques, M.SoftwareX2018,7, 1–5. 16 EOM- CCSD ΣPSD-I Δ (eV) ΣGW 17

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Förster, A.; Bruneval, F.J. Phys. Chem. Lett.2024,15, 12526–12534

2024

[2] [2]

Blase, X.; Duchemin, I.; Jacquemin, D.Chemical Society Reviews2018,47, 1022–1043

[3] [3]

Golze, D.; Dvorak, M.; Rinke, P.Frontiers in Chemistry2019,7, 377

[4] [4]

Bruneval, F.; Marques, M. A. L.Journal of Chemical Theory and Computation2013,9, 324–329

[5] [5]

Maggio, E.; Kresse, G.J. Chem. Theory Comput.2017,13, 4765–4778

2017

[6] [6]

Olsen,T.;Patrick,C.E.;Bates,J.E.;Ruzsinszky,A.;Thygesen,K.S.npj Computational Materials2019,5,106

[7] [7]

M.; Berkelbach, T

Lewis, A. M.; Berkelbach, T. C.Journal of Chemical Theory and Computation2019,15, 2925–2932

[8] [8]

J.; Whaley, K

Mejuto-Zaera, C.; Weng, G.; Romanova, M.; Cotton, S. J.; Whaley, K. B.; Tubman, N. M.; Vlček, V.Journal of Chemical Physics2021,154, 121101