{"id":"86f8d9aa-4428-4ef8-89f4-32601f14fc7e","arxiv_id":"2411.13167","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A new particle-particle Bethe-Salpeter kernel, expressed as a self-energy derivative, enables GW, T-matrix, and second-order approximations for double ionization potentials.","lead":"The paper derives a Bethe-Salpeter equation for the two-particle (particle-particle) channel using a pairing field and anomalous propagators, yielding approximate kernels for GW, T-matrix, and second-order self-energies. Benchmarks on 23 molecules show the GW and T-matrix kernels predict double ionization potentials with near-coupled-cluster accuracy.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The uncontrolled K→K0 linearization of the dynamical pp-BSE kernel (Eq. 30) is untested; if inaccurate, the TDA@dynBSE@GW conclusions shift, though the derivative-form kernel and static benchmarks stand.","rationale":"The paper's central claim is the derivative-form pp-BSE kernel, and the derivation of that formula appears sound. The numerical benchmarks are supporting evidence, and they split into static and dynamic parts. The static kernel results (ppBSE@GW, ppBSE@GT) do not involve the linearization, because the authors show that a static bare kernel leads to the same effective dynamic kernel. The only dynamical results, TDA@dynBSE@GW, are obtained through Eq. (30) after substituting K by K0. This is a separate, uncontrolled approximation: it is not derived from a small parameter, and the analogy with the eh-BSE does not automatically transfer because the structure of the noninteracting pp propagator differs. If the linearization is inaccurate, the reported dynamic corrections, the improved MAE, and the comparison with DIP-EOM-CCSD would shift. The reader's weakest_assumption identified exactly this step, and I agree that it is the most load-bearing concern. The proposed test is concrete and feasible because the code is open-source: iterating the kernel equation gives a direct estimate of the linearization error. If the test shows small deviations, the conditional verdict can be upgraded; if large, the dynamical benchmark would need revision. Because the reader already made the verdict conditional on this kind of sensitivity check, my assessment does not change the verdict.","tokens_in":34366,"tokens_out":5727,"duration_ms":64488,"concrete_test":"Using the public quack implementation, recompute TDA@dynBSE@GW for a representative subset (e.g., H2O, Ne, F2, CO) by iterating Eqs. (29)-(30): start with K=K0, build the dynamic kernel, solve the eigenvalue problem, use the resulting K to rebuild the kernel, and repeat until the DIPs change by less than 0.01 eV. Compare the converged DIPs to the linearized values in Table II; deviations larger than 0.1 eV would show that the K->K0 substitution is not benign. As a complementary check, vary the regularization eta in Eq. (54) over 0.01 to 0.5 Eh and report the resulting spread of the dynamic correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The formal result Xi_pp = delta Sigma^ee / delta G^ee appears internally consistent, and the static kernels (ppBSE@GW, ppBSE@GT) do not depend on the questionable step. The load-bearing weakness is the linearization introduced after Eq. (30): to make the frequency-dependent pp-BSE tractable, the effective kernel definition is evaluated by substituting K with K0. The authors justify this only by analogy with the eh-BSE ('might seem drastic but it has proven successful'), and no error estimate is provided. All dynamical numerical results, specifically the TDA@dynBSE@GW DIPs and the claim that the dynamical correction improves the MAE from 0.78 to 0.58 eV, are computed with this linearized kernel. Since K0 has poles at sums of two quasiparticle energies while the exact pp propagator has poles at DIPs and DEAs, the replacement can substantially alter the frequency denominators entering the dynamic correction; the approximation could be less benign in the pp channel than in the eh channel. This does not invalidate the central derivative formula, but it leaves the dynamical benchmark resting on an untested approximation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":34542,"tokens_out":7689,"duration_ms":79048,"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":[{"comment":"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.","section":"II.D, Eq. (30)"},{"comment":"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.","section":"II.E, Eqs. (35)-(38)"}],"minor_comments":[{"comment":"The sentence \"As one can readily seen from Eqs. (25) and (27)\" should read \"As one can readily see from Eqs. (25) and (27).\"","section":"II.C, after Eq. (27)"},{"comment":"The caption uses the label \"TDA@ppBSE@dynBSE\" while the main text and Table II use \"TDA@dynBSE@GW\"; please make the notation consistent.","section":"Figure 6 caption"},{"comment":"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.","section":"IV, Computational details"},{"comment":"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.","section":"V.A, Table I discussion"},{"comment":"The notation (K^{-1})(omega) and (K_0^{-1})(omega) is introduced without definition; please define these inverse kernels before first use.","section":"II.D, Eq. (30)"}],"recommendation":"major_revision","confidential_remarks":"The central derivative-form result and the static kernel benchmarks are sound and useful. The main risk is the K-to-K0 linearization in Eq. (30), which the authors themselves describe as 'drastic' and which underpins all dynamical results. If the authors validate the linearization or explicitly restrict their dynamical conclusions, I would be happy to support acceptance. There is no circularity or internal inconsistency in the formal derivation as far as I can see."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nNone 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.\n\nWho 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.","headline":"A genuinely new pp-BSE kernel formula with solid static benchmarks, but the dynamic correction rests on an untested K->K0 linearization.","tokens_in":35197,"tokens_out":1909,"would_cite":true,"duration_ms":20027,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A self-energy derivative gives the particle-particle Bethe-Salpeter kernel and practical double-ionization spectra.","keywords":["Bethe-Salpeter equation","particle-particle propagator","double ionization potential","anomalous propagator","pairing field","GW approximation","T-matrix approximation","GF(2) self-energy"],"falsifier":"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.","tokens_in":34071,"feed_emoji":"🧪","tokens_out":11261,"duration_ms":96444,"temperature":0.7,"pith_summary":"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.","feed_headline":"Double-ionization energies now from GW and T-matrix kernels","feed_subtitle":"A new pp-BSE kernel from anomalous propagators matches EOM-CCSD accuracy at lower cost","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Companion derivation of the anomalous-propagator response relations and the T-matrix anomalous self-energy that the pp-BSE kernel construction builds on.","marker":"[99]"},{"why":"Supplies the linearization procedure (replacing K by K0) that turns the frequency-dependent pp-BSE into a tractable eigenvalue problem.","marker":"[105]"},{"why":"Provides the perturbative dynamical-correction scheme used for the dynamic GW kernel.","marker":"[107]"},{"why":"Gives the second-order anomalous self-energy diagrams whose differentiation produces the GF(2) kernel.","marker":"[104]"},{"why":"Generalizes the self-energy formalism to anomalous propagators, underpinning the GW-type anomalous self-energy and its kernel.","marker":"[110]"},{"why":"Defines the particle-particle RPA (pairing-vibration approximation) that the first-order Coulomb kernel reproduces.","marker":"[57]"},{"why":"Supplies the 23-molecule valence benchmark set and the reference protocol from which FCI-quality DIPs are taken.","marker":"[114]"}],"fun_headline_variants":["Anomalous propagators make pp-BSE practical for DIPs","GW and T-matrix kernels: double ionization via pp-BSE","New pp-BSE kernels match EOM-CCSD accuracy on DIPs","T-matrix kernel leads in double ionization prediction","Pairing field paves way to accurate double ionization energies"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Anomalous propagators make pp-BSE practical for DIPs","GW and T-matrix kernels: double ionization via pp-BSE","New pp-BSE kernels match EOM-CCSD accuracy on DIPs","T-matrix kernel leads in double ionization prediction","Pairing field paves way to accurate double ionization energies"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0002,"raw_usage":{"total_tokens":1400,"prompt_tokens":996,"completion_tokens":404,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":612,"completion_tokens_details":{"reasoning_tokens":315}},"tokens_in":612,"tokens_out":404,"duration_ms":4875,"temperature":1.0,"reasoning_tokens":315,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:45:06.745139+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}