{"id":"5b1322d9-dac4-462e-b524-75a6681f0ad8","arxiv_id":"1908.01431","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"An NAO-based implementation of BSE in FHI-aims matches MolGW to about 1 meV and shows that the tier2+aug2 basis set is a cost-effective route to converged low-lying excitation energies.","lead":"This paper adds a new computational tool for predicting how molecules absorb light, using the Bethe-Salpeter equation inside the FHI-aims code. It reproduces existing calculations to about one millielectronvolt and proposes a small, cheap basis set that reaches nearly converged results.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The aug-cc-pV5Z reference that anchors the tier2+aug2 production-basis claim is not independently validated; if it carries residual basis error, the claim reduces to matching a reference rather than true CBS convergence.","rationale":"The reader identified the same load-bearing weak point: the aug-cc-pV5Z reference values used in Section IV.C are not independently established, and no residual basis-error estimate is given. This is the weakest premise because the production-basis recommendation depends on interpreting agreement with aug-cc-pV5Z as convergence to the complete basis set limit. The concern is real but not decisive: the paper's MolGW benchmark at TZVP is a controlled external test of the BSE implementation, the basis-set trends in Figs. 5, 6, and 8 are internally consistent, and the authors explicitly warn against systematic trust in the two-pole approximation in Section IV.B. No circularity, target-fitting, or post-hoc exclusion is evident. The appropriate remedy is a reproducibility check of the reference, not a rejection of the paper. Therefore the reader's ACCEPT verdict stands unchanged.","tokens_in":28975,"tokens_out":6824,"duration_ms":76339,"concrete_test":"Recompute BSE@G0W0@PBE for a representative subset (ethene, formaldehyde, pyrrole, naphthalene) using aug-cc-pV6Z, or at least extrapolate aug-cc-pVQZ/pV5Z/pV6Z, with the same G0W0 protocol as used in Section IV.C. If the lowest five singlet and triplet excitation energies shift by less than about 0.05 eV from aug-cc-pV5Z to aug-cc-pV6Z, and tier2+aug2 remains within about 0.1 eV of aug-cc-pV6Z, the production-basis claim is confirmed. As a cross-check, run one or two of these molecules with MolGW at aug-cc-pV5Z using identical G0W0 eigenvalues to verify that the reference values are code-independent.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central production claim is made in Section IV.C: the tier2+aug2 NAO basis yields lowest excitation energies within about 0.1 eV of aug-cc-pV5Z and is presented as effectively complete-basis-set converged for BSE (Figs. 5, 6, 8). The load-bearing premise is that aug-cc-pV5Z BSE values are themselves converged and trustworthy. The text never states which code produced the aug-cc-pV5Z reference values, nor does it quantify the residual basis error of that reference. Section IV.A validates the NAO BSE against MolGW only at the TZVP level and only with G0W0 quasiparticle eigenvalues supplied by MolGW; it does not independently validate the aug-cc-pV5Z values or the FHI-aims G0W0 protocol used in the convergence study. If aug-cc-pV5Z has finite-basis error of order 0.1 eV or more for some states, then the statement that tier2+aug2 is essentially at the complete-basis-set limit is not established. This is a provenance and residual-error gap, not an observed inconsistency. The 1 meV MolGW benchmark and the authors' explicit caveats about the two-pole self-energy in Section IV.B are genuine supporting evidence, so the concern does not overturn the acceptance, but it is the weakest load-bearing point in the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper presents an all-electron implementation of the Bethe-Salpeter equation (BSE) within the FHI-aims code using numeric atom-centered orbital (NAO) basis sets, following a G0W0@DFT starting point. The authors validate the BSE implementation against the MolGW package on Thiel's set of 28 molecules, using identical TZVP basis sets and MolGW-supplied G0W0 quasiparticle energies; they report mean absolute errors below 1 meV for the lowest ten singlet and triplet excitation energies, with and without the Tamm-Dancoff approximation. They then study basis-set convergence of BSE excitation energies for various NAO and Gaussian basis sets relative to an aug-cc-pV5Z reference, and propose a 'tier2+aug2' NAO basis that reproduces the reference within about 0.1 eV. The paper also investigates the effect of the G0W0 self-energy analytic continuation (two-pole vs 16-parameter Padé approximation), recommends an energy cutoff of 40 eV for unoccupied states, and compares BSE results with LR-TDDFT-LDA@PBE results.","tokens_in":29221,"tokens_out":14413,"duration_ms":135004,"significance":"If the results hold, this work provides a production-ready all-electron BSE capability in a widely used NAO-based code. The 1 meV agreement with MolGW under identical inputs is an exemplary controlled code-to-code validation, because it isolates the BSE matrix construction from uncertainties in the GW quasiparticle energies and basis sets. The basis-set analysis identifies a practical 'tier2+aug2' prescription that is much cheaper than aug-cc-pV5Z while yielding excitation energies within ~0.1 eV, which is valuable for routine applications. The discussion of self-energy analytic-continuation errors and the Ecut recommendation are also practically useful. The main weakness is the missing provenance of the aug-cc-pV5Z reference, which leaves the 'complete basis set limit' claim only partially supported.","major_comments":[{"comment":"The aug-cc-pV5Z reference values are not attributed to a specific electronic structure code, and the text does not state which G0W0 self-energy approximation (exact, two-pole, or 16-parameter Padé) was used to generate them. Because the basis-set convergence MAEs in Figs. 6 and 8 are computed against this reference, the reader cannot determine whether the ~0.1 eV agreement of tier2+aug2 reflects pure basis-set convergence or also differences in the GW self-energy treatment. Please specify the code and self-energy variant used for the reference, and ideally quantify the residual basis error of aug-cc-pV5Z by comparing with aug-cc-pVQZ or a larger basis. Without this, the Conclusions' claim that tier2+aug2 is 'essentially basis set converged' is not fully supported.","section":"§IV.C (Figs. 5, 6, 8)"},{"comment":"The paper recommends tier2+aug2 as essentially converged based on MAEs of about 0.1 eV against aug-cc-pV5Z, but Table I shows that the 16-parameter Padé self-energy approximation alone introduces MAEs of 0.06–0.08 eV relative to the exact MolGW self-energy at the TZVP level. Unless the same G0W0 eigenvalues are used for all NAO basis-set calculations and for the aug-cc-pV5Z reference, the 0.1 eV agreement cannot be attributed solely to basis-set completeness. The text should state explicitly whether the basis-set convergence results in Figs. 5, 6, and 8 use identical G0W0 input across all basis sets, and if not, the convergence claim should be reinterpreted accordingly.","section":"§IV.C and Table I"}],"minor_comments":[{"comment":"The last term in the integral should read 'L(62;5′2′)' rather than 'L(62;52′)' to match the double-primed index notation used elsewhere in the equation.","section":"Eq. (11)"},{"comment":"The word 'naphathalene' is misspelled; it should be 'naphthalene' in both table captions and in the accompanying text in §IV.F.","section":"Tables III and IV captions"},{"comment":"Several references (e.g., Refs. 55, 56, 57, 59, 60, 61, 62, 63, 65, 71) are missing journal names or full titles; please complete these entries for consistency and reproducibility.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically strong, and the 1 meV validation against MolGW is a convincing external benchmark. The major concerns above are about missing provenance and input consistency for the aug-cc-pV5Z reference, not about an observed numerical error. If the authors add a few clarifying sentences and, ideally, a small table showing aug-cc-pVQZ vs aug-cc-pV5Z differences, the paper would be suitable for acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a well-executed implementation paper, and the 1 meV agreement with MolGW is the real deal. The BSE implementation in FHI-aims is validated against an external code with identical inputs, so that part is solid. The tier2+aug2 basis recipe is a practical, useful contribution that others will adopt.\n\nWhat is genuinely new: first BSE implementation in an all-electron NAO code with this level of validation. The paper does the right thing by using identical G0W0 eigenvalues and TZVP basis in both codes for the benchmark, which isolates the BSE part. The basis set convergence study is careful, and the authors are honest about limitations - they explicitly warn that the two-pole error reduction is fortuitous cancellation, and they label the timings as unoptimized.\n\nWeak spots, in proportion. The weakest point is the aug-cc-pV5Z reference used for the convergence study. The text does not say which code produced those values, and it does not quantify the residual basis error of V5Z itself. The claim that tier2+aug2 is effectively complete-basis-set converged is therefore a match to a reference, not a proof of convergence to the true limit. That is a real caveat, but a minor one: for low-lying excitations, aug-cc-pV5Z is a credible reference, and the 1 meV MolGW agreement elsewhere gives confidence in the formalism. I would ask the authors to state the code and add a sentence on expected V5Z residual error, perhaps a V6Z spot check for one molecule.\n\nThe rest of the paper holds up. The Ecut analysis is useful, and the LR-TDDFT section is a reasonable add-on. No sign of circularity or fitting. This paper deserves a serious referee; I would accept after minor revision.","headline":"Well-executed validation of an NAO BSE implementation; the 1 meV MolGW agreement is genuine, and the tier2+aug2 recipe is useful, but the aug-cc-pV5Z reference provenance needs clarification.","tokens_in":29811,"tokens_out":2180,"would_cite":true,"duration_ms":22282,"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":"An atom-centered-orbital implementation of the Bethe-Salpeter equation reproduces reference excitation energies to 1 meV.","keywords":["Bethe-Salpeter equation","GW approximation","G0W0 quasiparticle energies","numeric atom-centered orbitals","basis set convergence","molecular excitation energies","resolution-of-identity approximation","linear-response TDDFT"],"falsifier":"Recompute the lowest five singlet and triplet BSE excitation energies for a subset of the benchmark molecules with an independent implementation and a still larger basis, such as aug-cc-pV6Z or a basis-set extrapolation from aug-cc-pVQZ and aug-cc-pV5Z; if these values shift by more than about 0.1 eV, the claim that tier2+aug2 is converged to the complete-basis-set limit would be disproven.","tokens_in":28732,"feed_emoji":"⚛️","tokens_out":12423,"duration_ms":119109,"temperature":0.7,"pith_summary":"This paper claims that neutral (optical) excitation energies of molecules can be computed accurately with the Bethe-Salpeter equation built on $G_0W_0$ quasiparticle energies, using numeric atom-centered orbital (NAO) basis sets instead of Gaussian orbitals. The paper implements the full three-step $G_0W_0$+BSE pipeline and tests it on a literature benchmark set of 28 small organic molecules. When both the new implementation and an established Gaussian-orbital reference code start from identical $G_0W_0$ quasiparticle eigenvalues and the same basis, the lowest ten singlet and triplet excitation energies agree to within 1 meV, and oscillator strengths to about $10^{-4}$. The paper further argues that a small augmented NAO basis, tier2+aug2, reproduces the much larger aug-cc-pV5Z reference values to about 0.1 eV across the whole set, making it a practical production basis. If these claims hold, molecular BSE calculations become routine at a cost comparable to ground-state density functional theory, with the same basis set serving both purposes.","feed_headline":"Excitation energies match reference to 1 meV","feed_subtitle":"An affordable augmented NAO basis reaches 0.1 eV of the large-basis reference, making BSE practical for molecules.","key_machinery":"The load-bearing object is the Bethe-Salpeter equation written as a matrix eigenvalue problem in electron-hole space, whose resonant block $A$ contains the $G_0W_0$ quasiparticle energy differences plus bare and statically screened Coulomb matrix elements, and whose coupling block $B$ contains the corresponding antiresonant terms. The implementation evaluates all Coulomb matrix elements through a resolution-of-identity (RI) expansion in auxiliary basis functions, reducing four-center integrals to sums of three-center coefficients and a dielectric matrix. Because only the zero-frequency value of the screened interaction enters the BSE kernel, the BSE step itself is independent of the analytic-continuation scheme used to obtain the $G_0W_0$ self-energy; the numerical tests in the paper explicitly show that the choice of that scheme affects only the input quasiparticle energies.","core_discovery":"The central discovery is that the electron-hole interaction part of the Bethe-Salpeter equation can be implemented on numeric atom-centered orbitals without losing numerical precision relative to established Gaussian-orbital implementations, provided the Coulomb and screened Coulomb integrals are handled by a resolution-of-identity expansion. Validated in this way, the implementation reproduces a reference code's lowest ten singlet and triplet excitation energies to 1 meV or better when the $G_0W_0$ input eigenvalues are identical, and reproduces oscillator strengths to about $10^{-4}$. The paper then isolates a separate question, basis-set convergence, and finds that un-augmented correlation-consistent bases converge slowly and can remain off by more than 0.5 eV even at quintuple-zeta level, while NAO bases augmented with two diffuse Gaussian functions reach about 0.1 eV of the aug-cc-pV5Z reference for low-lying excitations, with or without the antiresonant coupling block. The same convergence pattern holds for linear-response TDDFT with an adiabatic local-density kernel.","pith_inferences":["Editorial inference: the static BSE kernel depends on quasiparticle energies only through the matrix diagonal, so replacing single-shot $G_0W_0$ eigenvalues with more accurate self-consistent or contour-deformation values should transfer directly into BSE excitation energies, making affordable improvements in the GW step an immediate route to better optical gaps.","Editorial inference: the sensitivity of BSE to diffuse augmentation functions suggests that basis-set error in low-lying excitation energies is dominated by the description of unoccupied states and electron affinities; tier2+aug2 could serve as a cheap diagnostic for whether a given molecular basis is converged before engaging a larger reference.","Editorial inference: the 40 eV unoccupied-state cutoff is demonstrated for low-lying excitations, not for core-excited or high-lying states; a natural test is whether the same cutoff remains safe when the target orbitals are themselves high in energy."],"forward_implications":["If the implementation's 1 meV agreement with a reference code at fixed $G_0W_0$ input is generic, then the BSE step itself is numerically portable: any code supplying the same quasiparticle energies and basis will obtain the same neutral excitation energies to the meV scale.","The tier2+aug2 basis recipe places low-lying singlet and triplet excitation energies within about 0.1 eV of a much larger reference basis, so production BSE calculations can reuse the same NAO basis already used for ground-state DFT.","Truncating unoccupied states above a 40 eV cutoff changes low-lying excitation energies by about 10 meV while reducing the number of unoccupied states to about one third, with matrix memory savings close to an order of magnitude.","The basis-set convergence behavior demonstrated for BSE also holds for linear-response TDDFT with an adiabatic local-density kernel, so the tier2+aug2 prescription can be adopted for both methods.","The 16-parameter rational-continuation self-energy introduces mean errors near 0.1 eV in final BSE energies, roughly an order of magnitude larger than the 1 meV implementation error but smaller than the errors of the two-pole approximation."],"supporting_citations":[{"why":"defines the 28-molecule benchmark set, molecular geometries, and the best-estimate reference values whose comparison frames the paper.","marker":"[24]"},{"why":"provides the prior Gaussian-orbital BSE benchmark and quantifies the basis-set and starting-point errors that the present work reproduces and extends.","marker":"[43]"},{"why":"supplies the exact-real-axis $G_0W_0$ quasiparticle energies and the BSE reference values against which the present implementation is validated.","marker":"[73]"},{"why":"establishes the all-electron $G_0W_0$ and resolution-of-identity integral machinery on which the present BSE implementation builds.","marker":"[66]"},{"why":"defines the tiered numeric atom-centered orbital basis sets whose tier2 level is the core of the recommended tier2+aug2 recipe.","marker":"[82]"},{"why":"provides the correlation-consistent NAO basis sets used to demonstrate the slow convergence of un-augmented basis sets.","marker":"[83]"},{"why":"furnishes the shared TZVP basis in which the 1 meV agreement between implementations is established.","marker":"[109]"},{"why":"defines the aug-cc-pVnZ basis family, including the aug-cc-pV5Z reference used for all basis-set convergence comparisons.","marker":"[111]"},{"why":"formulates the two-pole analytic continuation whose influence on $G_0W_0$ eigenvalues and subsequent BSE energies is quantified.","marker":"[91]"},{"why":"formulates the 16-parameter rational-function continuation used as the more accurate alternative self-energy treatment.","marker":"[92]"}],"fun_headline_variants":["BSE on numeric orbitals hits 1 meV accuracy","Affordable basis set makes BSE practical for molecules","Numeric atom-centered orbitals match Gaussian BSE to 1 meV","New augmented NAO basis reaches 0.1 eV of large-basis limit","Electron-hole calculations on NAO basis reproduce reference to 1 meV"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The convergence claims depend on the largest reference basis (aug-cc-pV5Z) being effectively converged; if that reference itself carries hidden basis error, the 0.1 eV agreement of tier2+aug2 is only an agreement between two approximations, not convergence to the true limit.","fun_headline_variants_meta":{"raw":{"variants":["BSE on numeric orbitals hits 1 meV accuracy","Affordable basis set makes BSE practical for molecules","Numeric atom-centered orbitals match Gaussian BSE to 1 meV","New augmented NAO basis reaches 0.1 eV of large-basis limit","Electron-hole calculations on NAO basis reproduce reference to 1 meV"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000561,"raw_usage":{"total_tokens":2684,"prompt_tokens":988,"completion_tokens":1696,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":604,"completion_tokens_details":{"reasoning_tokens":1618}},"tokens_in":604,"tokens_out":1696,"duration_ms":13911,"temperature":1.0,"reasoning_tokens":1618,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:13:41.119484+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the lowest five singlet and triplet BSE excitation energies for a subset of the benchmark molecules with an independent implementation and a still larger basis, such as aug-cc-pV6Z or a basis-set extrapolation from aug-cc-pVQZ and aug-cc-pV5Z; if these values shift by more than about 0.1 eV, the claim that tier2+aug2 is converged to the complete-basis-set limit would be disproven.","supporting_citations":[],"review_version":1}