{"id":"5d114003-9d3c-45e2-b602-39f991c7fda1","arxiv_id":"2607.05853","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":5,"one_line_summary":"A GW+BSE implementation in the ABACUS+LibRPA framework uses real-space locality of the screened Coulomb interaction to Fourier-interpolate from a coarse to a dense k-mesh, validated by molecular and solid-state benchmarks.","lead":"The paper implements Bethe-Salpeter equation (BSE) calculations for optical spectra using numerical atomic orbitals and pseudopotentials, with a dual-k-mesh strategy that Fourier-interpolates the screened Coulomb interaction from a coarse to a dense k-grid. A smart generalist might read it because the dual-k-mesh approach reduces the computational cost of converged optical-excitation calculations in periodic systems, which is a known bottleneck in many-body perturbationtheory","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The dual-k-mesh strategy rests on W(R) truncation being safe, but this is only indirectly tested via cross-code benchmarks rather than isolated convergence verification.","rationale":"The reader correctly identified the real-space truncation of W(R) as the most load-bearing unverified assumption. This is the foundation of the dual-k-mesh strategy: if W(R) does not decay fast enough, the Fourier interpolation introduces aliasing errors that corrupt the BSE kernel on the dense k-mesh. The paper's benchmarks provide indirect evidence that the truncation works for the tested systems (Si, MgO, molecules), but they cannot isolate interpolation error from other sources of inter-code discrepancy. The concern does not invalidate the specific results presented — the agreement with FHI-aims, BerkeleyGW, VASP, and Exciting is convincing for these systems. Rather, it limits the generality of the 'reliable' claim: without an independent truncation convergence test or a clear delineation of where the assumption breaks down, the reader cannot know a priori whether the dual-k-mesh strategy is safe for a new material class. The paper is honest about other limitations (unconverged exciton binding energies, MgO intensity discrepancy, G0W0 gap underestimation), which supports the ACCEPT verdict. The concern is about the scope of the claim, not its correctness for the tested cases. The verdict remains ACCEPT because the implementation is validated for representative systems and the methodology is sound in principle; the missing truncation test is a gap in the validation rather than a flaw in the approach. The reader's HIGH confidence is slightly generous given this gap, but the overall assessment is fair.","tokens_in":30132,"tokens_out":2590,"duration_ms":222120,"concrete_test":"For Si or MgO, compute W_{μν}(q) directly on the dense 14×14×14 k-mesh (no interpolation) and compare the resulting BSE spectrum against the dual-k-mesh result (W computed on 7×7×7, interpolated to 14×14×14). If the spectra agree to within, say, 20 meV in peak positions and 5% in intensities, the truncation assumption is validated for these systems. Additionally, for a more stringent test, repeat with a system having a large dielectric constant (e.g., a small-gap semiconductor like InSb or a polar material with weak screening) and check whether the 7×7×7 supercell still suffices.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim — that Fourier interpolation of W_{μν}(R) from a coarse to a dense k-mesh is 'practical and reliable' — depends on W(R) decaying sufficiently fast within the Born–von Kármán supercell defined by the coarse k-mesh (Sec. III.B, step 1). The paper states that 'contributions from large |R| can be safely truncated' but provides no independent convergence test of the real-space truncation radius. The benchmarks (Si, MgO, Thiel's set) provide indirect evidence: if truncation were grossly wrong, spectra would not match other codes. However, these comparisons conflate multiple sources of discrepancy (basis set, frequency grid, q→0 treatment), so they cannot cleanly isolate the interpolation error. For the specific systems tested — moderate-gap semiconductors and small molecules — the truncation is likely safe given the screening. The concern is about the generality of the 'reliable' claim: for materials with large dielectric constants, small gaps, or weakly bound excitons (where W(R) is longer-ranged), the coarse-mesh supercell may be too small, introducing aliasing errors in the interpolated BSE kernel. The paper does not identify the class of materials where this could fail, nor does it provide a diagnostic for detecting such failures. This is the load-bearing assumption because if it fails, the entire efficiency advantage of the dual-k-mesh strategy is undermined — one would need to increase the coarse k-mesh, negating the cost savings.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"This manuscript presents a BSE implementation within the ABACUS+LibRPA framework using numerical atomic orbitals (NAOs) and norm-conserving pseudopotentials. The key methodological contribution is a dual-k-mesh strategy: the screened Coulomb interaction W is computed on a coarse k-mesh (inherited from the preceding G0W0 calculation), inverse-Fourier transformed to real space W_{μν}(R), and then Fourier-interpolated onto a dense k-mesh for BSE Hamiltonian assembly and diagonalization. The spatial locality of W(R) in the NAO+LRI representation makes this interpolation efficient. The paper includes convergence tests (NAO basis, auxiliary basis, k-point sampling with offset averaging), benchmarks against FHI-aims for Thiel's molecular set (28 molecules, MAD 32–44 meV), and cross-code comparisons for Si and MgO absorption spectra, plus exciton binding energies for six materials.","tokens_in":30241,"tokens_out":2642,"duration_ms":163004,"significance":"The dual-k-mesh strategy addresses a genuine computational bottleneck in BSE calculations for periodic systems. The implementation is cleanly specified, with careful treatment of Coulomb singularity (head/wing terms) and the truncated Coulomb operator. The use of velocity–length gauge equivalence as an internal diagnostic for basis-set incompleteness (Sec. IV.A.1) is a thoughtful contribution. The Thiel's set benchmark against FHI-aims, with identical Kohn–Sham starting points to isolate BSE-kernel differences, is well designed and yields quantitatively useful error bars. The cross-code benchmarks for Si and MgO span NAO, plane-wave, and LAPW frameworks, providing meaningful validation across fundamentally different numerical representations.","major_comments":[{"comment":"Sec. III.B, step 1: The statement that 'contributions from large |R| can be safely truncated' is the load-bearing assumption of the dual-k-mesh strategy, yet no independent convergence test of the real-space truncation radius for W(R) is provided. The cross-code benchmarks (Figs. 7–8, Table I) provide indirect evidence—if truncation were grossly wrong, spectra would not match—but these comparisons conflate multiple error sources (basis set, frequency grid, q→0 treatment). A brief test showing W(R) convergence with respect to the Born–von Kármán supercell size for at least one system (e.g., Si or MgO) would substantially strengthen the central claim. Alternatively, the authors should at minimum discuss the class of materials where slow decay of W(R) (large dielectric constant, small gap, weakly bound excitons) could compromise the interpolation, and note whether the coarse k-mesh used (7×","section":null},{"comment":"Table II: Several exciton binding energies are not converged with respect to k-mesh (e.g., GaN: 74→13 meV from 11×11×7 to 21×21×14; AlN: 85 meV at 20×20×12 vs. 147 meV converged value from Ref. [9]). The MgO value (284 meV) deviates substantially from experiment (80–145 meV). While the authors are transparent about this, the claim in the abstract that benchmarks 'collectively validate the accuracy' could be read as overstating the exciton binding energy results. The authors should clarify which values are converged and which are not, perhaps by marking unconverged entries in Table II.","section":null}],"minor_comments":[{"comment":"Sec. II.C, Eq. (20): The notation k_1 and k_2 for the two k-points in the BSE Hamiltonian is introduced without explicitly stating the momentum conservation condition k_2 = k_1 + q (or similar). A brief clarifying sentence would help readers unfamiliar with the convention.","section":null},{"comment":"Fig. 2: The workflow diagram is informative but dense. The distinction between 'Rmesh' and 'qmesh' could be made more explicit in the caption.","section":null},{"comment":"Sec. IV.A.3: The offset-averaging weights (1/27, 8/27, 6/27, 12/27) are stated without derivation. A one-line reference to the symmetry-reduction procedure would suffice.","section":null},{"comment":"Sec. IV.B.2, Fig. 8: The systematically lower peak intensities for MgO are attributed to 'differences in how high-lying unoccupied states and the associated transition matrix elements are represented.' This explanation is plausible but unverified. A brief comment on whether increasing the number of virtual orbitals was tested would be helpful.","section":null},{"comment":"Table I caption: '140 excitations per category' — it would be useful to note that this is 28 molecules × 5 excitations.","section":null},{"comment":"References [37–39] are cited as '2026' and appear to be very recent or in-press; please verify final publication details.","section":null},{"comment":"Sec. IV.B.1: The growth of discrepancy with conjugation length in polyenes is an interesting observation. The suggested explanation (delocalized exciton sampling larger kernel volume) is reasonable but speculative; a brief qualifier would be appropriate.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid implementation contribution that fits the journal's scope well. The W(R) truncation concern raised in review is legitimate but, on reading the paper, is not as severe as it might appear: the NAO+LRI framework's real-space locality is well-established in prior work (Zhou et al. [29], Ren et al. [48]), and the coarse k-meshes used (7×7×7) generate reasonably large supercells. The concern is primarily about the generality of the 'reliable' claim rather than a correctness issue for the tested systems. I recommend minor revision with a request for either a brief truncation test or a discussion of failure modes. The self-citation pattern (Refs. [35–39, 43–48]) is heavy but appropriate given that this is a cumulative implementation effort from the same group."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for the careful reading and constructive suggestions. Both major comments are well-taken. We will (1) add a dedicated W(R) real-space truncation convergence test for Si and MgO, and add a discussion of material classes where slow decay of W(R) could compromise the interpolation; (2) mark unconverged entries in Table II and soften the abstract language to avoid overstating the exciton binding energy results.","responses":[{"response":"The referee is correct that the real-space truncation of W(R) is the central assumption of the dual-k-mesh strategy and that the manuscript lacks a dedicated convergence test isolating this parameter. We will address this in two ways. First, we will add a convergence test showing the BSE absorption spectrum of Si (and, if space permits, MgO) computed with varying Born–von Kármán supercell sizes for W(R), while keeping all other parameters fixed. In our preliminary checks, the Si spectrum is already well converged when the supercell extent matches the 7×7×7 coarse k-mesh used for the G0W0 calculation, consistent with the fact that the cross-code benchmarks in Figs. 7–8 show no artifacts attributable to truncation. Second, we will add an explicit discussion of the class of materials where the spatial decay of W(R) could be slow—namely, systems with large static dielectric constants, small band gaps, or weakly bound (large-radius) excitons, such as narrow-gap semiconductors and highly polarizable metals. In such cases, the Born–von Kármán supercell inherited from the coarse k-mesh may be insufficient, and the user should verify convergence by increasing the supercell size. We note that for the systems studied in this work (Si, MgO, GaN, AlN, CdS, SnO2), the coarse k-mesh of 7×7×7 (or comparable) provides a supercell large enough that W(R) is effectively converged, as evidenced by the agreement with reference codes. We will state this explicitly in the revised manuscript.","revision_made":"yes","referee_comment":"Sec. III.B, step 1: The statement that 'contributions from large |R| can be safely truncated' is the load-bearing assumption of the dual-k-mesh strategy, yet no independent convergence test of the real-space truncation radius for W(R) is provided. [...] A brief test showing W(R) convergence with respect to the Born–von Kármán supercell size for at least one system (e.g., Si or MgO) would substantially strengthen the central claim. Alternatively, the authors should at minimum discuss the class of materials where slow decay of W(R) could compromise the interpolation, and note whether the coarse k-mesh used (7×...)"},{"response":"We agree with this assessment. Several entries in Table II are not fully converged with respect to k-mesh sampling, and the abstract language could be misread as claiming uniform validation across all benchmark types. We will make two changes. First, we will add a column or footnote to Table II explicitly marking which exciton binding energies are converged (Si and CdS are reasonably converged at 21×21×21) and which are not (GaN, AlN, MgO, and SnO2 still show significant k-mesh dependence). Second, we will revise the abstract to distinguish more carefully between the absorption spectrum benchmarks (where cross-code agreement is strong) and the exciton binding energies (where convergence is incomplete for several materials). Specifically, we will change 'collectively validate the accuracy of the present implementation' to language that acknowledges that the absorption spectra and molecular benchmarks validate the implementation, while the exciton binding energies are presented as a systematic study of k-mesh convergence behavior rather than as fully converged benchmark values. We believe this accurately reflects the content of the paper without overstating the results.","revision_made":"yes","referee_comment":"Table II: Several exciton binding energies are not converged with respect to k-mesh (e.g., GaN: 74→13 meV from 11×11×7 to 21×21×14; AlN: 85 meV at 20×20×12 vs. 147 meV converged value from Ref. [9]). The MgO value (284 meV) deviates substantially from experiment (80–145 meV). While the authors are transparent about this, the claim in the abstract that benchmarks 'collectively validate the accuracy' could be read as overstating the exciton binding energy results. The authors should clarify which values are converged and which are not, perhaps by marking unconverged entries in Table II."}],"tokens_in":29729,"tokens_out":1375,"duration_ms":58729,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"This is a solid computational methods paper. The main thing to know: it presents a BSE implementation within the ABACUS+LibRPA framework (pseudopotential + NAO basis) with a dual-k-mesh strategy that Fourier-interpolates the screened Coulomb interaction W from a coarse to a dense k-mesh. The formalism follows Zhou et al. [29] (all-electron FHI-aims), so the theoretical foundation is not new. What is new is the porting to a pseudopotential framework, the systematic convergence studies, and the practical offset-averaging scheme for accelerating spectral convergence.","headline":"Solid BSE implementation with dual-k-mesh interpolation; benchmarks are thorough, novelty is incremental, and the W(R) truncation assumption needs a direct convergence test but is probably fine for the systems studied.","tokens_in":30920,"tokens_out":701,"would_cite":true,"duration_ms":47690,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Fourier interpolation of screened Coulomb W cuts BSE k-mesh cost","keywords":[],"falsifier":"A material system where W_{μν}(R) decays so slowly that truncation on the Born-von Kármán supercell introduces errors comparable to or larger than the spectral features of interest, causing the interpolated BSE spectrum to diverge from a directly computed dense-k-mesh reference.","tokens_in":30310,"feed_emoji":"🔬","tokens_out":1222,"duration_ms":107607,"temperature":0.7,"pith_summary":"This paper presents an implementation of the Bethe-Salpeter equation (BSE) for optical excitation spectra using numerical atomic orbitals (NAOs) and norm-conserving pseudopotentials, integrated within the ABACUS+LibRPA software framework. The central innovation is a dual-k-mesh strategy: the computationally expensive screened Coulomb interaction W is computed on a coarse k-mesh (where GW quasiparticle energies are calculated), then Fourier-interpolated to an arbitrarily dense k-mesh for assembling and diagonalizing the BSE Hamiltonian. This interpolation is made possible by the localized resolution-of-identity (LRI) technique, which casts W into a real-space, unit-cell-indexed form W_{μν}(R) that is inherently short-ranged and can be truncated on a modest Born-von Kármán supercell. The paper validates this approach through systematic convergence tests (basis set, auxiliary basis, k-point sampling) and benchmark calculations on 28 molecules (Thiel's set) and two periodic solids (Si, MgO), demonstrating agreement with established codes at substantially reduced computational cost.","feed_headline":"Fourier interpolation of screened Coulomb W cuts BSE k-mesh cost","feed_subtitle":"Computing the screened interaction on a coarse k-mesh and interpolating to a dense one makes converged optical spectra affordable for solids","key_machinery":"The dual-k-mesh workflow: (1) DFT on both coarse and dense k-meshes; (2) GW calculation on coarse k-mesh producing quasiparticle energies and W_{μν}(R) in a small real-space supercell; (3) Fourier transform of W_{μν}(R) to dense k-mesh; (4) BSE Hamiltonian assembly and diagonalization on dense k-mesh. The LRI technique expands NAO products in atom-centered auxiliary basis functions, enabling the real-space representation. The velocity-length gauge equivalence serves as an internal diagnostic for basis-set incompleteness. Offset averaging over symmetry-reduced k-mesh shifts mitigates artifacts from high-symmetry points.","core_discovery":"The key finding is that the real-space screened Coulomb interaction W_{μν}(R), when expressed in the NAO+LRI framework, decays rapidly enough that it can be computed on a coarse k-mesh and Fourier-interpolated to a dense k-mesh without significant loss of accuracy in the resulting BSE absorption spectra. This separation of the k-mesh for W evaluation (coarse, affordable) from the k-mesh for BSE Hamiltonian construction (dense, converged) addresses the primary computational bottleneck in GW+BSE calculations for periodic systems. The paper demonstrates this with benchmarks showing mean absolute deviations of 32–44 meV against FHI-aims for molecular excitations, and consistent absorption peak结构","pith_inferences":["If W_{μν}(R) is sufficiently short-ranged for Fourier interpolation, the same dual-mesh strategy could apply to dynamical (frequency-dependent) screening beyond the static approximation used here, potentially improving exciton binding energy predictions for wide-gap materials like MgO and SnO2 where the paper reports large residuals (284 meV and 102 meV vs experiment).","The truncation of W at large |R| may break down for materials with large dielectric constants or weakly bound, spatially extended excitons (e.g., small-gap semiconductors near a dielectric catastrophe), where the screening cloud extends beyond the Born-von Kármán supercell. A quantitative truncation-error analysis as a function of dielectric constant would clarify the boundary of applicability.","The finding that BSE spectra are less sensitive to auxiliary basis completeness than GW self-energies suggests that a tiered basis strategy—using more complete auxiliary bases for GW and compressed bases for BSE—could yield further computational savings without spectral degradation."],"forward_implications":["The dual-k-mesh strategy makes converged BSE spectra for periodic systems accessible at a fraction of the uniform dense k-mesh cost, potentially enabling GW+BSE calculations for larger and more complex materials than previously feasible.","The real-space locality of W_{μν}(R) in the NAO+LRI framework could extend to other many-body quantities beyond BSE, such as finite-momentum exciton dispersion or nonlinear optical response, where dense k-sampling is similarly a bottleneck.","The systematic growth of code-to-code discrepancies with conjugation length in polyenes (up to ~200 meV for octatetraene triplet excitations) suggests that delocalized excitons stress the limits of localized basis representations and may motivate hybrid or adaptive basis strategies for extended molecular systems.","The offset-averaging technique for k-mesh artifacts provides a practical, parameter-free alternative to searching for optimal k-mesh shifts, generalizable to any BSE implementation suffering from high-symmetry-point artifacts."],"fun_headline_variants":["Dual-k-mesh approach lowers cost of BSE optical spectra for solids","Real-space W interpolation enables dual-k-mesh BSE calculations","Short-range W interpolation enables accurate low-cost BSE spectra","Interpolating screened Coulomb to a dense k-mesh cuts BSE cost","Dual-k-mesh BSE strategy avoids dense k-mesh cost for W evaluation"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The dual-k-mesh strategy depends on the screened Coulomb interaction W_{μν}(R) decaying fast enough in real space to be truncated on a small Born-von Kármán supercell. The paper states this truncation is safe but does not provide a quantitative analysis of truncation error versus system type, and for materials with large dielectric constants or weakly bound excitons, the range of W may be longer than assumed.","fun_headline_variants_meta":{"raw":{"variants":["Dual-k-mesh approach lowers cost of BSE optical spectra for solids","Real-space W interpolation enables dual-k-mesh BSE calculations","Short-range W interpolation enables accurate low-cost BSE spectra","Interpolating screened Coulomb to a dense k-mesh cuts BSE cost","Dual-k-mesh BSE strategy avoids dense k-mesh cost for W evaluation"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":1538,"prompt_tokens":590,"completion_tokens":948,"prompt_tokens_details":null},"tokens_in":590,"tokens_out":948,"duration_ms":30151,"temperature":1.0,"reasoning_tokens":839,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T22:21:38.029064+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"A material system where W_{μν}(R) decays so slowly that truncation on the Born-von Kármán supercell introduces errors comparable to or larger than the spectral features of interest, causing the interpolated BSE spectrum to diverge from a directly computed dense-k-mesh reference.","supporting_citations":[],"review_version":1}