Anomalous propagators and the particle-particle channel: Bethe-Salpeter equation

Antoine Marie; Pierre-Fran\c{c}ois Loos; Pina Romaniello; Xavier Blase

REVIEW 2 major objections 5 minor 2 cited by

Anomalous propagators and the particle-particle channel: Bethe-Salpeter equation

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A self-energy derivative gives the particle-particle Bethe-Salpeter kernel and practical double-ionization spectra.

desk verdict A genuinely new pp-BSE kernel formula with solid static benchmarks, but the dynamic correction rests on an untested K->K0 linearization. read the letter →

arxiv 2411.13167 v3 pith:KWW5VX7M submitted 2024-11-20 physics.chem-ph cond-mat.mtrl-scimath-phmath.MPnucl-th

Antoine Marie , Pina Romaniello , Xavier Blase , Pierre-François Loos This is my paper · ORCID

classification physics.chem-phcond-mat.mtrl-scimath-phmath.MPnucl-th

keywords Bethe-Salpeterequationparticle-particlepropagatordoubleionizationpotentialanomalouspairingfieldGWapproximationT-matrixGF(2)self-energy

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 derives a Bethe-Salpeter equation for the particle-particle (pp) propagator—the two-body Green's function whose poles are double ionization potentials (DIPs) and double electron affinities—in a form whose kernel is the derivative of an anomalous self-energy with respect to an anomalous propagator, exactly mirroring the familiar electron-hole BSE kernel. The derivation works through a pairing field that temporarily breaks particle-number conservation and through anomalous propagators, so the pp kernel is obtained by functional differentiation once a self-energy approximation is chosen. The authors construct pp-BSE kernels from the bare Coulomb interaction (recovering pp-RPA), from second-order and screened interactions (GF(2) and GW), and from the pp T-matrix, then benchmark the resulting DIPs on 23 small molecules against near-exact full configuration interaction references. The best kernel, based on the T-matrix effective interaction, reaches a mean absolute error of 0.45 eV for 46 valence singlet and triplet DIPs, competitive with DIP-EOM-CCSD at lower formal cost, and the GW kernel also improves double-core-hole energies. A sympathetic reader would care because this opens a systematic route to double-ionization and double-electron-attachment spectra analogous to the route that made GW-BSE standard for neutral excitations.

What carries the argument

The central objects are the anomalous propagators $G^{\mathrm{ee}}$ and $G^{\mathrm{hh}}$—pair propagators that vanish in a number-conserving ground state but respond to an external pairing field—and the pairing-field perturbation $\hat U^{\mathrm{pp}}$ that transiently breaks particle-number symmetry. The load-bearing identity is the Schwinger-type relation $K = \delta G^{\mathrm{ee}}/\delta U^{\mathrm{hh}}|_{U=0}$, which identifies the pp propagator as a linear response; differentiating the Gorkov-Dyson equation through this relation yields the pp-BSE with kernel $\Xi_{\mathrm{pp}} = \delta\Sigma^{\mathrm{ee}}/\delta G^{\mathrm{ee}}$. The second operative mechanism is the linearization of the frequency-dependent kernel by replacing the full propagator $K$ with the non-interacting $K_0$ inside Eq. (30), an approximation the paper imports from the electron-hole BSE literature to make the dynamic kernels numerically tractable.

What would settle it

Take a small molecule from the benchmark (e.g., H2O or N2) and solve the pp-BSE with the frequency-dependent kernel of Eq. (30) without the K-to-K0 substitution, iterating until the kernel's K dependence is self-consistent; if the resulting DIPs differ from the linearized values by more than the reported mean absolute error (0.45 to 0.78 eV depending on kernel), the linearization—not the derivative-form kernel—is doing the work.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central result is that the particle-particle Bethe-Salpeter kernel can be written as $\Xi_{\mathrm{pp}}(44';33') = \delta \Sigma^{\mathrm{ee}}(33') / \delta G^{\mathrm{ee}}(44')$ evaluated at vanishing pairing field, exactly the form of the electron-hole kernel $\Xi_{\mathrm{eh}} = \delta\Sigma/\delta G$. This turns the pp-BSE into a practical construction: choose an anomalous self-energy $\Sigma^{\mathrm{ee}}$, differentiate once with respect to the anomalous propagator $G^{\mathrm{ee}}$, and insert the resulting kernel into the Dyson equation whose poles are double ionization potentials (DIPs) and double electron affinities. The paper derives four such kernels—bare Coulomb (recovering pp-RPA), second-order/GF(2), GW-type screened interaction, and pp T-matrix—reduces the frequency-dependent equation to a non-Hermitian eigenvalue problem, and benchmarks it on 46 valence singlet and triplet DIPs of 23 molecules. The T-matrix kernel gives the lowest mean absolute error (0.45 eV), and the GW kernel with a perturbative dynamical correction reaches DIP-EOM-CCSD-level accuracy at a lower formal cost; the GW kernel also improves double-core-hole energies and supplies a first-principles justification for screening both Hartree and exchange terms in the pp interaction.

Load-bearing premise

The numerical DIPs are produced with Eq. (30) linearized by substituting the non-interacting propagator K0 for the full K inside the kernel, an approximation the paper itself calls drastic; if that substitution is inaccurate in the pp channel, the computed DIPs—and the benchmarking conclusions—shift.

Editorial extensions

If this is right

Every self-energy approximation with an anomalous counterpart now yields a pp-BSE kernel: bare Coulomb reproduces pp-RPA, second-order gives a GF(2) kernel, bubble-screened interactions give a GW kernel, and ladder-screened interactions give a T-matrix kernel.
The static GW kernel within the Tamm-Dancoff approximation gives DIPs with a mean absolute error of 0.78 eV at a much lower prefactor than DIP-EOM-CCSD, and adding the perturbative dynamical correction lowers the overall mean absolute error to 0.58 eV.
The static T-matrix kernel is the most accurate of the study, with a mean absolute error of 0.45 eV for both singlet and triplet DIPs, though it is more expensive because it requires the full effective-interaction tensor.
The GW kernel substantially improves double-core-hole energies with respect to pp-RPA, reducing errors from roughly 50 to 60 eV down to about 7 to 14 eV, a regime where linear-response methods and state-specific $\Delta$SCF remain complementary.
The same formalism applies in principle to double electron affinities, though the paper notes that systems binding two electrons are too large for the current implementation.

Reading between the lines

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

One step beyond the paper: the derivative-of-self-energy construction could be transferred to three-body propagator equations, whose kernels are currently built from bare Coulomb or ad hoc interactions; the pp-BSE W- and T-based kernels are natural candidates.
If the K-to-K0 linearization is as safe in the pp channel as in the eh channel, the frequency-dependent GW kernel could be iterated to self-consistency on a few molecules, giving a direct test of whether the dynamical correction saturates or oscillates.
The benchmark ordering (T-matrix more accurate than GW, which is more accurate than GF(2)) suggests that ladder diagrams carry much of the double-ionization correlation; a kernel combining ladder and bubble screening might outperform either alone.
The DEA sector of the same pp-BSE could be probed on spatially extended systems once active-space or Davidson-type solvers are ported from pp-RPA, connecting the present work to neutral-excitation calculations through (N±2) energy differences.

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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

The thing to know: this paper gives a clean, useful form for the particle-particle Bethe-Salpeter kernel as a functional derivative of the anomalous self-energy with respect to the anomalous Green's function, exactly analogous to the electron-hole case. That is genuinely new and it makes the pp channel tractable for systematic kernel approximations. The authors then derive GW, GF(2), and T-matrix kernels for the pp channel, and benchmark valence DIPs for 23 molecules against FCI-quality references. The static ppBSE@GT result, 0.45 eV MAE, is better than DIP-EOM-CCSD (0.61 eV) on this set; the dynamical GW correction (0.58 eV) is competitive. The derivation is careful, the SI is detailed, and the open-source implementation is a plus.

The main soft spot is the linearization in Eq. (30), where K inside the kernel is replaced by K0 to make the dynamic calculation tractable. The authors justify it by analogy to the eh-BSE, which 'might seem drastic,' but they provide no error estimate for the pp case. Since all the dynamic results, the TDA@dynBSE@GW numbers and the claimed improvement from 0.78 to 0.58 eV, rest on that substitution, this is a real gap. The static kernels do not depend on it, so the central derivative-form result and the static benchmarks survive even if the dynamic correction turns out to be less happy. The regularization eta=0.05 Eh is also hand-picked, and the TDA has to be enforced for BN and C2, which slightly mixes approximation levels.

None of this sinks the paper. The formal core is solid, the numerical study is honest, and the comparison with DIP-EOM-CCSD is fair. The dynamic correction claim would be stronger with a sensitivity analysis on eta and a direct test of the K->K0 approximation, but that is a request for revision, not a reason to reject.

Who gets value from this: anyone working with two-body Green's functions, double ionization spectra, or BSE kernels. It gives a practical recipe and clear benchmarks. I would send this to a serious referee; the referee should focus on the dynamic linearization and the regularization.

Referee Report

2 major / 5 minor

Summary. The manuscript presents a derivation of a Bethe-Salpeter equation for the two-body particle-particle propagator K using a pairing field and anomalous (Gorkov) propagators. The central formal result is Eq. (27), which expresses the pp kernel as Xi_pp = delta Sigma^ee / delta G^ee, in direct analogy with the electron-hole kernel Xi_eh = delta Sigma / delta G. From this expression the authors construct approximate kernels corresponding to the bare Coulomb (pp-RPA), second-order/GF(2), GW, and T-matrix self-energies, and they formulate the resulting pp-BSE as a non-Hermitian eigenvalue problem with optional Tamm-Dancoff and dynamical perturbative corrections. The numerical part benchmarks singlet and triplet valence DIPs for 23 molecules against FCI/CIPSI references: ppBSE@GW has MAE 0.66 eV, TDA@ppBSE@GW 0.78 eV, TDA@dynBSE@GW 0.58 eV, ppBSE@GF(2) 2.94 eV, and ppBSE@GT 0.45 eV, compared with DIP-EOM-CCSD at 0.61 eV. Single-site double-core-hole energies are also reported for four molecules and compared with CVS-FCI, DeltaSCF, and CVS-DIP-EOM-CCSD. The article concludes that the dynamic GW correction brings the pp-BSE to coupled-cluster accuracy and that the static T-matrix kernel is the most accurate of the tested kernels.

Significance. If correct, the derivative-form kernel in Eq. (27) is a useful unification: it gives pp-BSE the same constructive scheme as eh-BSE and provides a first-principles derivation of the screened pp kernel previously introduced ad hoc in the trion literature. The benchmark is carefully designed: the FCI reference values are extrapolated CIPSI numbers, the same 23-molecule set is used as in the authors' earlier IP benchmark, and the implementation is in an open-source code (quack). The TDA scalings and the O(N^6) comparisons with DIP-EOM-CCSD are stated explicitly, and the renormalization factors for the dynamical correction are tabulated. These are genuine strengths. The main limitation is that the dynamical results rest on an uncontrolled linearization of the effective kernel, so the numerical evidence for the dynamical correction is not yet as strong as the evidence for the static kernels.

major comments (2)

[II.D, Eq. (30)] The dynamic kernel used for all TDA@dynBSE@GW results is obtained by replacing K with K0 inside Eq. (30), with the justification being only an analogy to the eh channel. Because K0 has poles at sums of two quasiparticle energies while the exact pp propagator has poles at DIPs and DEAs, this substitution can alter the frequency denominators of the dynamic correction; since Table II's TDA@dynBSE@GW column and the claim that the dynamic correction lowers the MAE from 0.78 to 0.58 eV rest entirely on this linearized kernel, it is load-bearing and currently untested. Please add a numerical check of the linearization, for example by solving the nonlinear form iteratively for a subset of molecules or by comparing with an alternative treatment of the frequency dependence, or at minimum quantify the sensitivity of the reported dynamic shifts to the replacement.
[II.E, Eqs. (35)-(38)] The dynamical correction is computed as a first-order perturbative correction within the TDA, but the manuscript does not report any validation of this perturbative treatment in the pp channel. The smallest renormalization factor in Table II is 0.78, which is not very close to 1, so the first-order shift is not uniformly small. Since the entire TDA@dynBSE@GW column is produced by this procedure, please provide a convergence check (for example, a comparison with a direct nonlinear solution on small systems) or a discussion of why first-order perturbation theory is adequate despite renormalization factors significantly below unity.

minor comments (5)

[II.C, after Eq. (27)] The sentence "As one can readily seen from Eqs. (25) and (27)" should read "As one can readily see from Eqs. (25) and (27)."
[Figure 6 caption] The caption uses the label "TDA@ppBSE@dynBSE" while the main text and Table II use "TDA@dynBSE@GW"; please make the notation consistent.
[IV, Computational details] Please state explicitly that the value eta = 0.05 Eh is used to regularize the diverging denominators in Eq. (54), and report whether the dynamical corrections are sensitive to this choice.
[V.A, Table I discussion] The text states that the MAE and MSE associated with ppRPA@HF are both 2.95 eV, which averages the singlet and triplet entries (2.89 and 3.02 eV); this averaging should be stated explicitly to avoid an apparent mismatch with the table.
[II.D, Eq. (30)] The notation (K^{-1})(omega) and (K_0^{-1})(omega) is introduced without definition; please define these inverse kernels before first use.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the pp-BSE kernel derivation and benchmarks are self-contained; the K→K0 linearization is a stated limitation, not a circular step.

full rationale

The central result is the derivation of the pp-BSE with kernel Xi_pp = delta Sigma^ee / delta G^ee (Eqs. (26)-(27)). This is obtained by functional differentiation of the anomalous self-energy with respect to the anomalous propagator, starting from the response relation K = delta G^ee / delta U^hh (Eq. (17)) and the Gorkov-Dyson equation (Eq. (23)). Although Eqs. (17), (22), and (23) are attributed to the authors' prior work (Ref. 99), the paper states that 'a fully detailed derivation is provided in the Supplementary Material,' so the self-citation is not load-bearing in the sense of hiding an unstated assumption. The approximate kernels (Coulomb, GF(2), GW, T-matrix) are each obtained by differentiating an explicit self-energy ansatz of the form Sigma^ee = -i W G^ee; the resulting matrix elements are new algebraic expressions and are not fitted to the DIPs being predicted. The numerical benchmarks compare the resulting DIPs against independent FCI/CIPSI and DIP-EOM-CCSD references, with no adjustable parameter tuned to those targets. The one significant caveat is the linearization after Eq. (30): 'Equation (30) is thus linearized by substituting K by K0. This approximation might seem drastic but it has proven successful in the eh-BSE case.' This is an acknowledged uncontrolled approximation affecting the dynamical correction, but it is not a circular reduction: the predicted DIPs are not equal to any input by construction. It is a limitation to weigh in assessing the dynamical results, not a circularity in the derivation. Overall, the derivation chain is self-contained and the predictions are externally benchmarked, so no significant circularity is found.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on the Gorkov/Nambu formalism and standard many-body perturbation theory. The axioms above are the domain assumptions specific to this paper. No new particles or entities are postulated.

free parameters (1)

regularization parameter eta = 0.05 Eh
Used in the perturbative dynamic correction to regularize the denominators of Eq. (54). The value is chosen by hand; no sensitivity analysis is reported, and the smallest renormalization factor (0.78) is used to argue the perturbation series is well-behaved.

assumptions (4)

domain assumption Anomalous propagators vanish at U=0 for number-conserving systems, so only one term survives in the derivative chain rule leading to Eq. (26).
Section II C, after Eq. (26): 'only one of them is non-zero for a number-conserving system (i.e., at U=0)'.
ad hoc to paper The linearized kernel (replace K by K0 in Eq. (30)) is a valid approximation.
Section II D: 'Equation (30) is thus linearized by substituting K by K0. This approximation might seem drastic but it has proven successful in the eh-BSE case.' This is load-bearing because all numerical results use it.
domain assumption The approximate Sigma^ee must contain exactly one anomalous propagator G^ee for a non-zero kernel.
Section II C: 'According to the definition of Xi_pp in Eq. (27), the approximate expression of Sigma^ee must contain exactly one anomalous propagator G^ee for the derivative to be non-zero.' This restricts which diagrams contribute.
domain assumption One-body Green's functions from one-shot GW, GF(2), and T-matrix approximations (linearized quasiparticle equation) provide adequate starting points.
Section IV: the underlying one-body Green's function calculations rely on the linearized quasiparticle equation without self-consistency. The starting-point dependence is tested numerically, but the approximations themselves are not derived in this paper.

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Pith. "Pith review of Anomalous propagators and the particle-particle channel: Bethe-Salpeter equation." pith.science (2026). https://pith.science/paper/KWW5VX7M

@misc{pith2026241113167,
  author       = {Pith},
  title        = {Pith review of: Anomalous propagators and the particle-particle channel: Bethe-Salpeter equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KWW5VX7M}},
  note         = {Machine review of arXiv:2411.13167}
}

abstract

The Bethe-Salpeter equation has been extensively employed to compute the two-body electron-hole propagator and its poles which correspond to the neutral excitation energies of the system. Through a different time-ordering, the two-body Green's function can also describe the propagation of two electrons or two holes. The corresponding poles are the double ionization potentials and double electron affinities of the system. In this work, a Bethe-Salpeter equation for the two-body particle-particle propagator is derived within the linear-response formalism using a pairing field and anomalous propagators. This framework allows us to compute kernels corresponding to different self-energy approximations ($GW$, $T$-matrix, and second-Born) as in the usual electron-hole case. The performance of these various kernels is gauged for singlet and triplet valence double ionization potentials using a set of 23 small molecules. The description of double core hole states is also analyzed.

Figures

Figures reproduced from arXiv: 2411.13167 by the authors.

**Figure 2.** FIG. 2. Diagrammatic representation of the eh-BSE, as defined in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Diagrammatic representation of the pp-BSE [see Eq. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4. Diagrammatic representation of the two direct second [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. Histogram of the errors (with respect to FCI) for the singlet and triplet principal DIP of 23 small molecules computed [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6. Histogram of the errors (with respect to FCI) for the singlet and triplet principal DIP of 23 small molecules computed in [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 8.** Figure 8: FIG. 8. Errors (with respect to CVS-FCI) associated with [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 7.** Figure 7: FIG. 7. Histogram of the errors (with respect to FCI) for [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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A new coupled-cluster method that includes four-hole-two-particle excitations and triple clusters gives near-exact double ionization potentials for small molecules.
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cond-mat.str-el 2025-01 conditional novelty 6.0 of 10

The (4,0)-multichannel Dyson equation, coupling two- and four-body Green's functions, describes single and double neutral excitations in a two-level model, unlike static BSE.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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However, for notational consistency, we prefer the terminology eh-RPA in the present context

note The aforementioned RPA formalism is usually referred to as particle-hole (ph) RPA. However, for notational consistency, we prefer the terminology eh-RPA in the present context. Stop

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Lu , author Y

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doi:10.1103/physrevlett.102.206411