{"id":"646f89e3-9dab-426f-a59f-1c5167888cd2","arxiv_id":"2505.06021","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A localized-orbital periodic RPA implementation with a dual k-grid and damped Coulomb potential reaches fast thermodynamic-limit convergence and reproduces the benchmark RPA@PBE CO/MgO adsorption energy.","lead":"This paper reports an implementation of the random phase approximation (RPA) in the BAND/AMS code using localized atomic orbitals, pair-atomic density fitting, a dual reciprocal-space grid, and k-grid-dependent Coulomb damping. It demonstrates fast convergence for 2D systems and gives a RPA@PBE adsorption energy of -1.67 ± 0.31 kcal/mol for CO on MgO(001), matching earlier RPA@PBE results.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Final ±0.31 kcal/mol omits documented projector-threshold and auxiliary-basis sensitivities; the CO/MgO agreement rides on error cancellation.","rationale":"The paper is a serious methods contribution. The RPA/ACFDT formalism, PADF expansion, dual-grid construction, and parallelization are described in enough detail to reproduce the algorithm, and the convergence tests in Figs. 4, 6, 7 and 12 support the claim of fast k-grid convergence for the tested systems. The final CO/MgO number also matches an independent periodic RPA@PBE calculation, which is real evidence in favor of the implementation. The load-bearing weakness is not the formalism but the honesty of the uncertainty budget for the headline number. The manuscript itself documents large absolute sensitivities (Table 4: ~17 kcal/mol in subsystem correlation energies when eps_d changes) and an incomplete auxiliary-basis convergence at QZ4P (Table 3: 0.20 kcal/mol shift from T3 to T4), and Sec. 4.5 explicitly warns that the projector introduces a subsystem-dependent effect beyond standard BSSE. Eq. (34) nevertheless quotes an error bar built only from the CBS fit intercept and the coverage correction, with no component for these effects. Because the agreement with Bajdich et al. is a difference of 0.02 kcal/mol, the meaningfulness of that agreement depends entirely on cancellation being stable. That is a concrete, testable condition, not a stylistic objection. If the requested recomputation shows the omitted terms are below 0.1 kcal/mol, the current claim stands; if not, the error bar and the 'excellent agreement' wording need revision. This is the same concern the reader identified, so the CONDITIONAL verdict is appropriate.","tokens_in":29015,"tokens_out":16690,"duration_ms":167200,"concrete_test":"Repeat the Table 4 protocol at eps_d = 5e-4 and 1e-4 with the T3 and T4 auxiliary bases for all three subsystems, and re-run the three-point CBS fit in Fig. 11 and Eq. (33)-(34) using these consistently recomputed values. Propagate the spread in ΔEc as a systematic uncertainty (including the QZ4P T3-to-T4 auxiliary shift of 0.20 kcal/mol). If the final adsorption energy shifts by more than 0.31 kcal/mol, the stated error bar and the agreement claim are not supported; if it remains within that band, the cancellation is robust under the tested variations.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central numerical claim depends on cancellation of large systematic errors that the final uncertainty budget does not include. Table 4 shows that changing the projector threshold eps_d from 1e-3 to 5e-4 shifts the absolute RPA correlation energies of MgO-CO, CO, and MgO by about 17 kcal/mol, while the adsorption difference only moves by 0.02 kcal/mol. Table 3 shows the QZ4P auxiliary-basis result is not fully converged even at T4 (ΔEc changes by 0.20 kcal/mol between T3 and T4), and the text in Sec. 4.5 states that the number of functions projected out in each subsystem depends on the other subsystem's basis, adding an effect beyond conventional BSSE. The final error in Eq. (34) combines only the CBS-interpolation error (0.29) and the coverage correction (0.12), explicitly neglecting layer and k-grid terms and omitting these documented projector/auxiliary sensitivities. The agreement with Bajdich et al. (-1.67 vs -1.65 kcal/mol) is within 0.02, so if the cancellation weakens for other basis sets or systems, the 'excellent agreement' and the proposed error bar are not supported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reports a periodic implementation of the RPA correlation energy in the BAND module of AMS, using numerical atomic orbitals and pair-atomic density fitting (PADF). The core methodological contributions are a Fermi-Dirac damping of the Coulomb potential whose range is tied to the k-grid ('AUTO damping'), a dual reciprocal-space grid scheme that adds points near the Gamma point to accelerate convergence to the thermodynamic limit, and a parallelization strategy based on splitting the (q, omega) integration over separate ScaLapack contexts. The implementation is tested on h-BN monolayers and bilayers, on fluoro-polyacetylene, and on CO adsorbed on MgO(001). For the latter, the authors report a counterpoise-corrected RPA@PBE adsorption energy of -1.67 +/- 0.31 kcal/mol in the low-coverage limit, which they describe as in excellent agreement with the earlier periodic RPA@PBE value of Bajdich et al. (-1.65 kcal/mol) and as less negative than experiment and embedded CCSD(T) results.","tokens_in":29193,"tokens_out":5406,"duration_ms":54617,"significance":"If the claims hold, the paper provides a useful and efficient canonical RPA implementation for two-dimensional and slab systems in a localized basis, with three notable strengths: (i) the k-grid convergence is rapid and shown to be nearly independent of the primary basis set, which is practically valuable; (ii) the AUTO damping criterion removes a user-tuned parameter for the Coulomb truncation, and the dual-grid scheme demonstrably reduces the Gamma-point sampling error; and (iii) the parallelization benchmark shows near-perfect strong scaling to thousands of cores, which is a practical advantage for production calculations. The convergence tests are systematic, spanning k-grids, damping radii, primary and auxiliary basis sets, projector thresholds, and coverage. However, the final application to CO/MgO carries a headline uncertainty that omits several documented sources of error, and the CBS extrapolation rests on a three-point linear fit; these issues affect the reliability of the stated final result but do not invalidate the methodological contribution.","major_comments":[{"comment":"The uncertainty budget for the final adsorption energy is incomplete. The text states that 'the only source of error in this number stems from the basis set extrapolation' and combines only the CBS intercept error (0.29 kcal/mol) and the coverage correction error (0.12 kcal/mol). However, Table 4 shows that changing the projector threshold epsilon_d from 1e-3 to 5e-4 shifts the RPA contribution to the adsorption energy by 0.02 kcal/mol, while going to 1e-4 shifts it by 0.18 kcal/mol (from -7.53 to -7.35 kcal/mol). Table 3 shows the QZ4P auxiliary-basis result is not fully converged even at T4, with the relative correlation energy changing by 0.20 kcal/mol between T3 and T4. These documented, quantifiable sensitivities are not included in the reported +/- 0.31 kcal/mol error bar, so the stated uncertainty under-represents the known uncertainty of the final number.","section":"Sec. 4.6.3, Eq. (34)"},{"comment":"The complete-basis extrapolation is based on a linear fit of the RPA adsorption-energy contribution against the inverse number of basis functions using only three points (DZP, TZ2P, QZ4P). The extrapolated value (-9.25 +/- 0.29 kcal/mol) lies 0.89 kcal/mol below the largest-basis computed value (-8.36 kcal/mol), and the reported uncertainty is the intercept standard error of that three-point fit. This does not account for the model error of the assumed 1/N_bas linear form, nor for the residual auxiliary-basis incompleteness at the QZ4P/T4 point. The authors should either provide additional evidence for the extrapolation law (e.g., a four-point fit or a second functional form) or enlarge the error bar to reflect the sensitivity of the intercept to the choice of fitting points.","section":"Sec. 4.5.1 and Fig. 11"},{"comment":"The paper correctly acknowledges that the projector method introduces an effect beyond conventional BSSE: the number of primary basis functions projected out in a subsystem depends on whether the other subsystem's basis is present. This means the counterpoise correction of Eq. (29) may not fully remove the basis-set superposition effect. The magnitude of this effect is, however, not quantified or bounded anywhere in the manuscript, and it is not included in the final uncertainty. Given that the final adsorption energy is a delicate balance between large absolute correlation energies (about 17 kcal/mol shifts in Table 4), the authors should estimate this contribution, for example by comparing projector thresholds in the dimer and monomer calculations, or at least justify why it is negligible at the CBS limit.","section":"Sec. 4.5, text near Eqs. (29)-(30)"}],"minor_comments":[{"comment":"The abstract and conclusions describe the agreement with Bajdich et al. as 'excellent', while the introduction says 'good agreement'. The quantitative basis for the word 'excellent' should be stated or the wording made consistent.","section":"Abstract and Conclusions"},{"comment":"In the figure caption, 'with respect to ts values calculated with a 17 x 17 k-mesh' appears to contain a typo; it should likely read 'its values calculated with a 17 x 17 k-mesh'.","section":"Sec. 4.3, Fig. 6 caption"},{"comment":"The definition of the cumulative projector in Eq. (21) is unclear because the R(k) matrices are defined in the k-dependent eigenbases of S(k). Please specify the common basis in which the matrix product over k is taken, or clarify how the k-dependent matrices are combined.","section":"Sec. 2.3, Eqs. (20)-(22)"},{"comment":"The sentence 'the TZ2P results are adjusted for the resulting fit error of 0.14 kcal/mol' does not specify whether this adjustment is applied to the TZ2P entries in Table 3 or to the CBS extrapolation in Fig. 11. Please state where the adjustment enters.","section":"Sec. 4.5.1, paragraph on TZ2P adjustment"},{"comment":"The table does not report the k-grid used for the 50% and 25% coverage calculations. For reproducibility, the k-grid for each coverage entry should be stated.","section":"Sec. 4.6.2, Table 5"},{"comment":"The meaning of the terms EQZ4P_HF, EQZ4P_PBE, and EQZ4P_XC@PBE in Eq. (33) should be defined explicitly, in particular whether these are total energies or adsorption-energy contributions and how the signs are chosen.","section":"Sec. 4.6.3, Eq. (33)"}],"recommendation":"major_revision","confidential_remarks":"The methodological core of the paper, the dual-grid scheme and the AUTO damping, appears sound and well tested, and the parallel-scaling results are convincing. The main weakness is the final application: the error bar in Eq. (34) excludes documented projector-threshold and auxiliary-basis sensitivities, and the CBS extrapolation is based on a three-point linear fit. These are fixable with additional estimates or revised text, but they are load-bearing for the headline adsorption energy. The paper is within the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the implementation work is solid: the dual k-grid with automatic Fermi-Dirac damping gives fast, basis-set-independent k-grid convergence for 2D systems, and the parallel scaling is genuinely good. Second, the final CO/MgO adsorption energy with its ±0.31 kcal/mol error bar is more fragile than the abstract suggests, because the quoted uncertainty excludes documented auxiliary-basis and projector-threshold sensitivities.\n\nWhat is actually new is the specific combination of dual reciprocal-space grids (an independent Q-grid augmenting the regular K-grid, inspired by the staggered-mesh method) with k-grid-dependent Coulomb damping and PADF in a canonical RPA code, validated on h-BN and CO/MgO. The ingredients are not new, and the authors credit the staggered-mesh and damped-Coulomb literature; the contribution is the integration and the demonstrated convergence behavior. The parallelization over (q,ω) pairs with separate ScaLapack contexts is a practical engineering solution that works, as the scaling data show.\n\nThe soft spot is the error budget on the application. Table 4 shows that changing the projector threshold ε_d from 1e-3 to 5e-4 shifts absolute RPA correlation energies by roughly 17 kcal/mol in the combined system while the adsorption difference moves by only 0.02 kcal/mol. That is a lot of cancellation to rely on without a quantified uncertainty term. Table 3 shows the QZ4P auxiliary-basis series is not converged even at T4 (0.20 kcal/mol change from T3 to T4), and Sec. 4.5 itself notes that the projector removes different numbers of functions depending on the other subsystem's basis, an effect beyond conventional BSSE. The final error in Eq. (34) combines only the CBS-interpolation error (0.29) and the coverage correction (0.12), explicitly dismissing layer and k-grid errors and omitting these documented sensitivities. So the agreement with Bajdich (−1.67 vs −1.65) is a helpful sanity check, but the claimed error bar likely understates the real uncertainty.\n\nThere is also no public code, data, or input files, which makes independent verification hard, and the conclusion says a complete account of the algorithm and detailed benchmarks will appear in future work — a hint that this manuscript is a partial report.\n\nWho this is for: anyone doing periodic RPA with localized basis sets, especially for molecule-surface and 2D systems. The central algorithmic claims hold up, so it deserves a serious referee and likely publication after revision. I would send it to review, with the instruction that the authors either broaden the uncertainty budget or soften the error-bar claim, and ideally release input files for the key numbers. The discussion of error cancellation alone is worth having in the literature.","headline":"Solid NAO-based periodic RPA implementation with fast k-grid convergence; the headline CO/MgO error bar omits documented sensitivities and the Bajdich agreement rides on error cancellation.","tokens_in":29801,"tokens_out":5632,"would_cite":true,"duration_ms":46587,"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 dual reciprocal-space grid makes periodic RPA correlation energies converge rapidly for two-dimensional materials.","keywords":["random phase approximation","periodic RPA","numerical atomic orbitals","pair-atomic density fitting","dual k-grid scheme","Coulomb damping","adsorption energy","MgO(001)"],"falsifier":"Repeat the CO/MgO(001) calculation at 100% coverage with the projector threshold lowered from $10^{-3}$ to $5\\times 10^{-4}$ while increasing the auxiliary basis until absolute RPA correlation energies stop shifting; if the RPA contribution to the adsorption energy moves by substantially more than the quoted 0.31 kcal/mol uncertainty, the error cancellation underlying the projector result breaks.","tokens_in":1758,"feed_emoji":"⚡️","tokens_out":2884,"duration_ms":126236,"temperature":0.7,"pith_summary":"The paper reports a practical implementation of the random phase approximation (RPA) for periodic systems using a localized numerical atomic orbital basis with pair-atomic density fitting. Its central claim is that a dual reciprocal-space grid—a regular momentum-transfer grid augmented near the $\\Gamma$-point—combined with a k-grid-dependent damping of the Coulomb potential, makes RPA correlation energies converge quickly and reliably to the thermodynamic limit for 2D systems. For CO on MgO(001), the implementation yields an RPA@PBE adsorption energy of $-1.67 \\pm 0.31$ kcal/mol after basis-set and coverage extrapolations, close to a previous periodic RPA value, while being less negative than experimental and CCSD(T) results, as expected for RPA@PBE. If the claim holds, it gives researchers a route to affordable canonical RPA calculations for molecule–surface interactions in a localized basis, with controlled convergence and good parallel scaling.","feed_headline":"Dual k-grids make periodic RPA converge fast for 2D materials","feed_subtitle":"Numerical-orbital RPA reaches the thermodynamic limit on small k-grids and matches earlier CO-on-MgO results.","key_machinery":"The load-bearing object is the dual grid pair $(K_G, Q_G)$: the momentum-transfer grid $Q_G$ is a regular sampling with the $\\Gamma$ point removed and extra points inserted at small displacements around it, while the k-grid $K_G$ is built so that all differences $k_j - k_l$ lie in $Q_G$. This keeps the computational cost of the polarizability proportional to the product of the two grid sizes while making memory growth linear in the number of added points. The second mechanism is a Fermi-Dirac damped Coulomb potential, with radius $r_0$ chosen automatically from the real-space supercell imposed by the k-sampling, which regularizes the $\\Gamma$-point divergence without introducing an artificial short-range truncation. The third piece is a periodic projector that removes basis combinations whose overlap eigenvalues fall below a threshold, keeping the pair-atomic density fitting expansion stable.","core_discovery":"The paper's central claim is that the slow k-grid convergence of RPA correlation energies in periodic systems, caused by the Coulomb singularity at $q = \\Gamma$, can be overcome by a dual-grid sampling scheme. A regular momentum-transfer grid has the $\\Gamma$ point removed and a few extra points inserted close to it, and the k-grid is constructed so that differences of k-points cover this enhanced grid. Combined with an automatic Fermi-Dirac damping of the Coulomb potential whose range grows with the k-grid, this yields RPA correlation energies that converge with small meshes, independent of basis set. For CO on MgO(001) at the RPA@PBE level, the final low-coverage adsorption energy is $-1.67 \\pm 0.31$ kcal/mol, matching a previous periodic RPA result while underestimating both experiment and recent embedded CCSD(T) values, as RPA@PBE is known to do.","pith_inferences":["The reported sensitivity of absolute correlation energies to the projector threshold suggests that the final adsorption-energy accuracy leans on error cancellation between subsystems; for systems where this cancellation is less complete, total energies may carry a larger unquantified error than the quoted $\\pm 0.31$ kcal/mol.","The dual-grid construction is not tied to RPA: the same $\\Gamma$-point treatment could accelerate periodic MP2 correlation energies or GW self-energies in localized-basis codes, where analogous k-grid convergence problems appear.","Because damping is tied to the k-grid rather than to a fixed physical cutoff, the method's converged absolute energies are range-separated intermediates; direct comparisons of absolute RPA correlation energies between codes with different k-grids may be misleading even when energy differences converge.","If RPA@PBE0 (hybrid input orbitals) indeed makes adsorption energies more negative, the implementation's natural next step—combining it with self-consistent periodic exact exchange—would directly test whether the gap to experiment and CCSD(T) is an input-orbital issue or an RPA kernel issue."],"forward_implications":["RPA correlation energies for 2D slabs and monolayers can be converged to the thermodynamic limit with modest k-meshes; the paper reports $9\\times 9$ grids giving adsorption energies converged within 0.02–0.05 kcal/mol.","k-point convergence is nearly independent of the one-particle basis, so practitioners can converge the k-grid with a small basis and then extrapolate the basis at a fixed coarse k-grid.","The parallelization over q and frequency pairs achieves near-perfect strong scaling to thousands of cores, removing the wall-clock bottleneck for canonical RPA on small-unit-cell surface calculations.","At RPA@PBE, CO on MgO(001) binds by $-1.67 \\pm 0.31$ kcal/mol in the low-coverage limit, confirming the known tendency of RPA@PBE to underestimate this adsorption energy relative to experiment and CCSD(T).","The same machinery—dual grids, automatic damping, and pair-atomic density fitting—is a direct base for more advanced ACFDT correlation kernels such as $\\sigma$-functionals or renormalized adiabatic kernels."],"supporting_citations":[{"why":"It supplies the pair-atomic density fitting framework and the projector method that the periodic implementation generalizes.","marker":"[123]"},{"why":"It provides the long-range Coulomb damping idea in periodic Fock exchange that the automatic Fermi-Dirac damping adapts.","marker":"[134]"},{"why":"It introduces the staggered-mesh sampling that the dual k-grid scheme is inspired by.","marker":"[135]"},{"why":"It extends the staggered mesh to the RPA and ring coupled cluster doubles, motivating the $\\Gamma$-point treatment used here.","marker":"[136]"},{"why":"It gives the previous periodic RPA@PBE adsorption energy used as the reference value for the final result.","marker":"[137]"},{"why":"It demonstrates local-RI RPA for periodic systems with numerical atomic orbitals, the context for the PADF-based implementation.","marker":"[89]"},{"why":"It defines the pair-atomic resolution of identity and the auxiliary basis sets that the correlation calculations rely on.","marker":"[126]"},{"why":"It proposes truncating the Coulomb potential beyond the real-space grid imposed by the k-sampling, the basis for the automatic damping.","marker":"[138]"},{"why":"It provides the experimental and embedded CCSD(T) adsorption energies against which the RPA@PBE result is judged.","marker":"[35]"}],"fun_headline_variants":["Dual k-grids fast-track periodic RPA to convergence","RPA gets a k-grid boost for 2D materials","Faster periodic RPA with dual reciprocal grids","Dual-grid RPA converges quickly, matches CO on MgO","New RPA scheme speeds up 2D surface adsorption"],"cache_read_input_tokens":31872,"weakest_assumption_plain":"The calculation rests on the assumption that discarding basis combinations whose overlap eigenvalues fall below a fixed threshold removes only redundant directions, so that the large shifts in absolute correlation energies caused by the threshold cancel between the molecule and the surface.","fun_headline_variants_meta":{"raw":{"variants":["Dual k-grids fast-track periodic RPA to convergence","RPA gets a k-grid boost for 2D materials","Faster periodic RPA with dual reciprocal grids","Dual-grid RPA converges quickly, matches CO on MgO","New RPA scheme speeds up 2D surface adsorption"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000408,"raw_usage":{"total_tokens":2096,"prompt_tokens":904,"completion_tokens":1192,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":520,"completion_tokens_details":{"reasoning_tokens":1109}},"tokens_in":520,"tokens_out":1192,"duration_ms":7447,"temperature":1.0,"reasoning_tokens":1109,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:49:59.140699+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the CO/MgO(001) calculation at 100% coverage with the projector threshold lowered from $10^{-3}$ to $5\\times 10^{-4}$ while increasing the auxiliary basis until absolute RPA correlation energies stop shifting; if the RPA contribution to the adsorption energy moves by substantially more than the quoted 0.31 kcal/mol uncertainty, the error cancellation underlying the projector result breaks.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the pair-atomic density fitting framework and the projector method that the periodic implementation generalizes."},{"cited_title":"M.; Pauly, F","cited_arxiv_id":null,"evidence_quote":"It provides the long-range Coulomb damping idea in periodic Fock exchange that the automatic Fermi-Dirac damping adapts."},{"cited_title":"Staggered Mesh Method for Correlation Energy Calculations of Solids: Second-Order Møller-Plesset Perturbation Theory","cited_arxiv_id":null,"evidence_quote":"It introduces the staggered-mesh sampling that the dual k-grid scheme is inspired by."},{"cited_title":"Staggered Mesh Method for Correlation Energy Calculations of Solids: Random Phase Approximation in Direct Ring Coupled Cluster Doubles and Adiabatic Connection Formalisms","cited_arxiv_id":null,"evidence_quote":"It extends the staggered mesh to the RPA and ring coupled cluster doubles, motivating the $\\Gamma$-point treatment used here."},{"cited_title":"K.; Vojvodic, A","cited_arxiv_id":null,"evidence_quote":"It gives the previous periodic RPA@PBE adsorption energy used as the reference value for the final result."},{"cited_title":"A Quadratic Pair Atomic Resolution of the Identity Based SOS-AO-MP2 Algorithm Using Slater Type Orbitals","cited_arxiv_id":null,"evidence_quote":"It defines the pair-atomic resolution of identity and the auxiliary basis sets that the correlation calculations rely on."},{"cited_title":"Efficient calculation of the exact exchange energy in periodic systems using a truncated Coulomb potential","cited_arxiv_id":null,"evidence_quote":"It proposes truncating the Coulomb potential beyond the real-space grid imposed by the k-sampling, the basis for the automatic damping."}],"review_version":1}