{"id":"c113e387-22d0-4d06-8dc5-54dcf1769a92","arxiv_id":"2607.20728","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A scalable plane-wave implementation of analytical BSE excited-state forces, built on a Z-vector Lagrangian and low-rank dielectric screening, enables BSE-relaxed geometries in hundreds-atom supercells and identifies where BSE changes TDDFT relaxation.","lead":"The authors implemented analytical forces for excited states described by the Bethe–Salpeter equation in a plane-wave code, making structural relaxation of hundreds-atom defect supercells feasible. The method is used to show when BSE and TDDFT agree (NV− in diamond) and when BSE is needed (carbon-dimer defect in 2D h-BN).","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Fixed-scissor and frozen-W approximations mean the forces are not full BSE-G0W0 derivatives; the paper explicitly acknowledges and bounds this, so the accepted verdict stands with the same scope caveat.","rationale":"I read the paper as claiming an approximate but practical BSE-level force engine, not a fully self-consistent BSE-G0W0 gradient. The central claim has multiple independent supports: analytical vs finite-difference agreement at the same approximation level (Table SII), reproduction of qualitative and semi-quantitative small-molecule geometries against full BSE-G0W0 and CC3/CCSDR(3) references (Sec. SIII), and the CO PEC tests. The paper's own Fig. 3 is the most honest stress test: it shows exactly where the approximation fails (PBE in hBN) and where it is controlled (DDHα). The reader's weakest-assumption call aligns with mine. I would not reject or conditionally accept because the authors do not conceal the approximation and the method's scope is explicitly limited by the need for a hybrid-like starting point. The one check I would want before relying on the method for a new inhomogeneous system is a residual-force comparison at the optimized geometry against full BSE-G0W0, which is the concrete test above.","tokens_in":24825,"tokens_out":10499,"duration_ms":112488,"concrete_test":"Compute full BSE-G0W0 analytical forces (or finite-difference forces from BSE-G0W0 PECs along several independent modes) at the fixed-scissor BSE-DDHα optimized ES geometry of CBCN in 2D-hBN, using the same or a smaller cluster/supercell model. If the largest residual force on the defect atoms exceeds the paper's 0.03 eV/Å relaxation threshold, the optimized geometry is not stationary on the true BSE-G0W0 PES and the fixed-scissor approximation would be biasing the headline relaxation result. If residuals fall below threshold, the approximation is validated for the decisive application.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest load-bearing point is that the analytical forces are exact derivatives of a surrogate BSE Hamiltonian, not of the full BSE-G0W0 functional. In evaluating U, the paper sets δH^QP/δ⟨φ_v| ≈ δH^KS/δ⟨φ_v| (fixed scissor) and δW/δ⟨φ_v| ≈ 0 (SM S1, Eq. S12). Thus any geometry optimization finds a stationary point for a geometry-independent QP correction and a W whose orbital response is frozen; it is not necessarily a stationary point of the BSE-G0W0 PES. The CO test (End Matter) shows this is benign near the relaxed bond length, but it also notes larger deviations at shorter bond lengths. Fig. 3 is the key evidence: for the PBE starting point in 2D-hBN the fixed-scissor and BSE-G0W0 PECs differ substantially, so the method's reliability is not intrinsic—it depends on the starting point having small, geometry-independent QP corrections. Since the paper's flagship inhomogeneous-screening application (CBCN in hBN) is exactly a case where screening and QP corrections are nontrivial, the DDHα starting point is doing load-bearing work. This is a real limitation, but it is explicitly stated, tested, and scoped in the discussion; it does not invalidate the implementation or the applied conclusions, provided the method is used within that regime.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This Letter reports a plane-wave implementation of analytical nuclear forces for BSE excited states, built on density-matrix perturbation theory, a Z-vector Lagrangian, the projective dielectric eigenpotential (PDEP) representation of screening, and GPU acceleration. The method requires a single response equation rather than 3N_atom Sternheimer solves, and it avoids explicit empty-state sums. The authors validate the implementation against finite-difference forces for the NV- center in diamond with PBE, DDH, and HSE starting points, against small-molecule BSE geometries from a fully analytical molecular implementation, and against finite-difference derivatives for CO under the same approximations. They then apply the method to NV- in diamond and the carbon-dimer defect C_BC_N in 2D h-BN, comparing TDDFT and BSE relaxed geometries and photoluminescence spectra. The two central approximations—a fixed scissor for the quasiparticle Hamiltonian derivative and neglect of the orbital derivative of W—are stated explicitly and are the subject of the CO test and the PEC comparison in Fig. 3.","tokens_in":25190,"tokens_out":6486,"duration_ms":61599,"significance":"The paper makes a substantial methodological contribution: it demonstrates a scalable path to BSE-level excited-state geometry relaxation in condensed systems, a capability that has so far been limited to small molecules. The implementation is carefully validated, with analytical finite-difference force agreement at the ~10^-5 Ry/bohr level, molecular benchmarks against fully analytical BSE-G0W0 forces, and an explicit test of the fixed-scissor approximation in the end matter and Fig. 3. The two applications are meaningful: for NV- in diamond the comparison with hybrid-TDDFT delineates where simpler methods suffice, while for the 2D h-BN defect the BSE result identifies the qualitative failure of semilocal TDDFT and the importance of screened electron-hole attraction. The authors are transparent about the surrogate nature of the forces—they are derivatives of a BSE Hamiltonian with a fixed scissor and frozen W, not of the full BSE-G0W0 functional—and they provide evidence on the conditions under which this surrogate is reliable.","major_comments":[],"minor_comments":[{"comment":"The title and abstract describe the method as 'analytical forces from the Bethe-Salpeter equation' without qualification. Since the forces are derivatives of a BSE Hamiltonian with a fixed-scissor approximation for QP derivatives and with δW/δ<φ_v| set to zero, I recommend adding a qualifier such as 'with fixed-scissor/frozen-W approximations' in the abstract (and at least a clarifying sentence in the introduction) to avoid readers interpreting the forces as full BSE-G0W0 gradients. The paper itself already states the scope in the theoretical framework, but the abstract currently overstates it.","section":"Title/Abstract"},{"comment":"The notation dω/dR in Eq. (4) should be explicitly defined as the derivative at fixed KS orbitals; otherwise the reader may not immediately see that the Z-vector formalism is needed because the orbitals are geometry-dependent. A one-sentence clarification would help.","section":"Theoretical framework, Eq. (4)-(7)"},{"comment":"The BSE VEEs are fitted by construction: the scissor is chosen to reproduce the BSE-G0W0 VEE at the ground-state geometry. The AEEs are therefore not fully independent predictions. The text notes this for VEEs, but it would be useful to add a footnote to both tables stating that the BSE AEEs inherit the geometry-independent scissor fitted at the GS geometry, so the apparent agreement with experiment is a combined test of the BSE model and the scissor approximation.","section":"Tables I and II"},{"comment":"The potential energy curve test of the fixed-scissor approximation is performed for only one interpolation coordinate in a 128-atom supercell, while the production CBCN calculations use a 288-atom supercell. The authors should state whether the smaller cell changes the conclusions, and ideally report the maximum deviation between fixed-scissor and BSE-G0W0 forces or AEEs along the tested path. This would make the 'reliable for DDHα' conclusion more quantitative.","section":"Fig. 3 and discussion"},{"comment":"There is a typo: 'for the save of computational cost' should read 'for the sake of computational cost.' ","section":"SM S4"},{"comment":"The sentence 'at a cost only a few times larger than the corresponding TDDFT forces calculations for hybrid-functional starting points' is supported for DDH (~9x for NV-, ~4x for hBN) but not for PBE (~73x for NV-). Since the sentence already limits the claim to hybrid starting points, it may be worth adding the PBE comparison explicitly to avoid an unintended implication that BSE-PBE is similarly close to TDDFT-PBE.","section":"Table SIII and main text"},{"comment":"The statement 'consistent with the absence of an explicit screened electron-hole attraction in TDDFT with semilocal functionals' is useful. Consider also stating that the DDHα TDDFT result includes an analogous effective attraction through the exact-exchange term, which is why TDDFT-DDHα and BSE-DDHα differential densities are nearly identical. This helps the reader understand why the starting point, not just the BSE kernel, is the controlling factor.","section":"SM S5, Fig. S2/S3"}],"recommendation":"minor_revision","confidential_remarks":"The paper fits the journal well and the technical content is convincing. My only substantive concern is terminological: 'analytical BSE forces' in the title may be read as full BSE-G0W0 gradients, whereas the implementation is explicitly a fixed-scissor/frozen-W surrogate. The authors' own Fig. 3 and CO validation show that they understand and test this scope, so I consider this a presentation issue rather than a flaw. The empty major-comment list reflects that the central methodology is sound and the limitations are made transparent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid methods letter, and referees should engage. The genuinely new piece is the plane-wave periodic Z-vector implementation of analytical BSE forces, with PDEP low-rank screening, no empty-state sums, and no 3N Sternheimer solves. The two defect applications (NV− in diamond, 511 atoms; CBCN in 2D hBN, 288 atoms) are exactly the scale that was previously out of reach, and the cost table makes the improvement concrete.\n\nThe validation is done right. Analytical forces match finite-difference forces to ~10^-5 Ry/bohr for PBE, DDH, and HSE starting points. The small-molecule benchmarks against fully analytical BSE-G0W0 geometries (Tölle et al.) check the physics, not just the code. The SM gives the Z-vector equations and the response kernels in enough detail to reproduce the implementation.\n\nThe soft spot is the one the authors flag themselves: the forces are not full derivatives of the BSE-G0W0 energy. The fixed-scissor and frozen-W approximations are in Eqs. (S9) and (S12). The paper does three things right about this. It tests the approximations on CO and shows they degrade at short bond lengths. It shows in Fig. 3 that for a PBE starting point in 2D hBN the fixed-scissor PEC deviates substantially from BSE-G0W0, while for DDHα the two PECs agree. It then builds the hBN conclusion on the DDHα starting point. So the limitation is load-bearing but openly scoped and actually demonstrated. What could have been a hidden flaw is instead a stated regime of validity.\n\nTwo minor things. The PL spectra are aligned to the ZPL, so the comparison is about line shape, not absolute position; that's standard and fine, but it puts more weight on the computed Huang–Rhys factors. Also the BSE forces cost is given in GPU hours on A100s; useful, though a convergence study in the PDEP number for forces would be good, not just energies.\n\nOverall: the central claim holds up. The method is a real advance for the field, and the paper is honest about where it does not apply. I would send it to referees.","headline":"Solid methods letter: plane-wave BSE forces with an honest, well-tested scissor/frozen-W scope; the fixed-scissor caveat is real but the paper does the work to show when it holds.","tokens_in":25639,"tokens_out":1816,"would_cite":true,"duration_ms":18950,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["71.15.-m","71.35.-y"],"model":"deepseek-v4-flash","headline":"An analytical-force implementation of the Bethe-Salpeter equation replaces one response calculation per atomic motion with a single Z-vector equation, bringing excited-state relaxation to defect supercells of hundreds of atoms.","keywords":["Bethe-Salpeter equation","excited-state forces","analytical nuclear gradients","Z-vector equation","density-matrix perturbation theory","projective dielectric eigenpotential","point defects in semiconductors","photoluminescence line shapes"],"falsifier":"Relax the carbon-dimer defect in 2D h-BN from a PBE starting point using fully analytical BSE-G0W0 forces, available in Gaussian-basis implementations, and compare with the fixed-scissor BSE relaxation: the paper's Fig. 3 already shows the two potential-energy curves diverging, so the G0W0-relaxed geometry would differ measurably in bond lengths and photoluminescence sideband. Applying the same comparison to any system with a strongly geometry-dependent self-energy correction at a hybrid starting point would delimit where the approximations bind.","tokens_in":24688,"feed_emoji":"⚛️","tokens_out":16188,"duration_ms":112373,"temperature":0.7,"pith_summary":"The paper's goal is to make structural relaxation of electronic excited states affordable at the Bethe-Salpeter equation (BSE) level for solids, by computing analytical forces rather than differentiating energies numerically. The key algorithmic claim is that a Z-vector Lagrangian combined with a low-rank representation of the screened Coulomb interaction reduces the cost from one response calculation per nuclear displacement to a single equation, with no explicit empty-state sums, so that GPU-accelerated supercells of several hundred atoms become feasible. On two point defects, the paper argues when that machinery changes the answer: for the nitrogen-vacancy center in diamond, BSE and hybrid-functional TDDFT agree closely, while for the carbon-dimer defect in two-dimensional h-BN, only the screened electron-hole attraction in BSE stabilizes the localized defect excitation and corrects a relaxation pattern that semilocal TDDFT gets qualitatively wrong. A reader would care because these forces control photoluminescence line shapes, self-trapped excitons, and the behavior of defect-based quantum emitters — quantities previously out of reach of BSE-level theory for realistic supercells.","feed_headline":"Excited-state relaxation now reaches 500-atom supercells","feed_subtitle":"One response equation replaces per-atom solves; screened electron-hole attraction fixes a 2D defect TDDFT misses.","key_machinery":"Central object: the Z-vector (Handy-Schaefer) response equation — one Sternheimer-like solve for orbitals Z_v whose source U is the derivative of the BSE excitation energy with respect to occupied KS orbitals. It yields the relaxed density Δρ(z); with the unrelaxed Δρ(x) from BSE amplitudes, dω/dR = ∫ (∂V_ext/∂R)(Δρ(x)+Δρ(z)), replacing 3N per-atom DFPT solves. Companion mechanism: the PDEP (projective dielectric eigenpotential) low-rank form of the screened interaction W, avoiding full dielectric matrices and empty-state sums. Two explicit approximations carry it: a fixed scissor shift in place of the QP-Hamiltonian derivative, and δW/δ⟨φ_v|≈0. Wannier localization, an inexact Krylov solver","core_discovery":"Central claim: the derivative of a BSE excitation energy is the integral of the external-potential derivative against a differential density built from BSE response orbitals plus Z-vector orbitals from one Handy-Schaefer response equation, at a cost only a few times that of TDDFT forces. After validation, the method relaxes the triplet 3E state of NV- in a 511-atom diamond supercell and the first singlet state of the CBCN defect in a 288-atom 2D-hBN supercell. For NV-, BSE and TDDFT give nearly identical displacements and photoluminescence spectra; for CBCN, only BSE with a dielectric-dependent hybrid starting point reproduces experiment, the screened electron-hole interaction keeping the re","pith_inferences":["The same Lagrangian machinery should extend to analytical non-adiabatic couplings and excited-state vibrational line widths, since the Z-vector source U already contains the orbital-response information those derivatives require; the paper lists both as future work.","A practical screening rule follows from the two defect case studies: start with hybrid-functional TDDFT forces and escalate to BSE forces only when the dielectric environment is strongly inhomogeneous or the excited state delocalizes — a policy directly applicable to computational searches for quantum emitters.","The paper's Fig. 3 pattern implies a precise prediction: the scissor approximation will fail exactly where the self-energy correction is large and geometry-dependent, for example a defect near an interface or surface even at a hybrid starting point; testing such a system would sharpen the method's domain of validity.","Because the frozen-W approximation fixes screening at the ground-state geometry, the forces inherit a residual starting-point dependence that only a self-consistent GW workflow would remove; until then, differences in relaxed geometry between starting points may rival the differences between TDDFT and BSE themselves."],"forward_implications":["Excited-state geometry relaxation at the BSE level becomes routine for defect supercells of hundreds of atoms; the reported cost is only a few times that of hybrid-functional TDDFT forces (hundreds of GPU-hours per force evaluation).","For localized deep defects in quasi-homogeneous hosts such as NV- in diamond, hybrid-functional TDDFT is a reliable and much cheaper surrogate: BSE and TDDFT agree on displacements and photoluminescence line shapes within the starting-point dependence.","For defects in inhomogeneous dielectric environments such as the carbon dimer in 2D h-BN, semilocal TDDFT relaxes toward an excited state of the wrong character; the screened electron-hole attraction of BSE is required to keep the excitation localized.","The fixed-scissor approximation used in the forces is reliable only when the starting point leaves a small, geometry-independent quasiparticle correction; a dielectric-dependent hybrid starting point restores that condition, whereas a semilocal starting point does not (the paper's Fig. 3)."],"fun_headline_variants":["BSE forces scale to 500-atom supercells","One response equation replaces per-atom solves","BSE fixes relaxation TDDFT gets wrong","Screened electron-hole attraction corrects 2D defect"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the quasiparticle correction behaves as a fixed, geometry-independent scissor shift and that screening barely changes when occupied orbitals move; where this fails, the forces are not derivatives of the actual BSE energy surface — a breakdown the paper's own Fig. 3 shows for the PBE starting point in 2D h-BN.","fun_headline_variants_meta":{"raw":{"variants":["BSE forces scale to 500-atom supercells","One response equation replaces per-atom solves","BSE fixes relaxation TDDFT gets wrong","Screened electron-hole attraction corrects 2D defect"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000722,"raw_usage":{"total_tokens":3067,"prompt_tokens":725,"completion_tokens":2342,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":469,"completion_tokens_details":{"reasoning_tokens":2288}},"tokens_in":469,"tokens_out":2342,"duration_ms":13454,"temperature":1.0,"reasoning_tokens":2288,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T09:31:07.416125+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Relax the carbon-dimer defect in 2D h-BN from a PBE starting point using fully analytical BSE-G0W0 forces, available in Gaussian-basis implementations, and compare with the fixed-scissor BSE relaxation: the paper's Fig. 3 already shows the two potential-energy curves diverging, so the G0W0-relaxed geometry would differ measurably in bond lengths and photoluminescence sideband. Applying the same comparison to any system with a strongly geometry-dependent self-energy correction at a hybrid starting point would delimit where the approximations bind.","supporting_citations":[],"review_version":1}