{"id":"2405ad54-6199-4e59-8496-41afe1ad6e4d","arxiv_id":"2411.18493","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A deep-learning-inspired ADAM optimizer for orbital rotations plus a softmax parameterization for occupation numbers extends natural orbital functional calculations to 1000 electrons, the largest yet.","lead":"This paper adapts the ADAM optimizer from deep learning to speed up natural orbital functional calculations, enabling a 1000-electron hydrogen cluster to be simulated. The work could make a promising quantum chemistry method practical for much larger strongly correlated systems than before.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The scaling claim lacks convergence diagnostics: the 1000-electron PNOF7 runs are stopped by a schedule, not by a stationarity test, so smooth potential curves do not establish that ADAM reached a true minimum.","rationale":"The reader's weakest assumption was the reliability of PNOF7 at 1000-electron scale, with no comparison to systematically improvable methods. I agree that this is a real limitation, but the more load-bearing condition for the paper's central claim is optimization convergence itself. The paper's headline advance is that the deep-learning-inspired optimizer makes large NOF calculations feasible; if the 1000-electron run is not demonstrably at a stationary point, the scale claim collapses regardless of PNOF7's intrinsic accuracy. The manuscript provides no convergence diagnostics: no gradient norms, no energy-vs-iteration curves for the large systems, no restart tests, and no comparison against established NOF optimizers on smaller cells. Algorithm 1's early-stopping rule is a schedule, not a stationarity test. This is an internal-evidence gap, not a disagreement with consensus, so it should be settled by a concrete numerical test. The paper is otherwise plausible: the ADAM adaptation is clearly described, hyperparameters are specified, and the softmax parameterization for occupations is a sensible choice. If the proposed test shows agreement with reference optimizers on smaller systems and initialization-independent energies at scale, the conditional acceptance can be upgraded. For now, the conditional verdict remains appropriate.","tokens_in":13451,"tokens_out":3757,"duration_ms":38206,"concrete_test":"Recompute the 10x10x10 hydrogen cube dissociation curve at r = 1.0, 2.0, 2.46, 3.0, and 4.0 Å with a strict convergence test, e.g., max |g_pq| < 10^-6 and energy change < 10^-8 Eh, using two independent initial orbital/occupation guesses. Compare these energies with Fig. 1a. Separately, run PNOF7 on H8 and H27 cubes with both the new ADAM scheme and the previous iterative-diagonalization or trust-region optimizer. If the ADAM energies differ from the reference minima by more than 1 mEh, or if the two initial guesses converge to different energies at large r, the convergence claim is not supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the ADAM-based orbital-rotation scheme achieves convergent NOF calculations on unprecedentedly large systems, with the 1000-electron hydrogen cube as the flagship. Algorithm 1 in the Supplementary Information specifies early stopping when the energy fails to improve, followed by learning-rate reduction and an increased iteration budget, but it never defines a convergence threshold on the gradient norm or on energy changes. Consequently, the outer loop can terminate at a non-stationary point, halted by a bookkeeping condition rather than by evidence of a minimum. This is not academic: at stretched H-H distances the PNOF7 energy landscape has near-degenerate orbitals and many flat directions, and the AMSGrad-style max second moment in Eq. (11) monotonically grows, which can shrink all step sizes and stall progress before convergence. The smooth dissociation curves in Fig. 1 cannot distinguish a converged minimum from a slow crawl along a flat valley, especially because no iteration counts, gradient norms, or restart checks are reported for the 1000-electron run. Even if PNOF7 were exact for this system, unconverged stationary points would invalidate both the 'largest NOF calculation' claim and the quantitative estimate of the critical distance rc. This concern is distinct from, and logically prior to, the reader's concern about PNOF7 accuracy at scale: the calculation must first be shown to be converged before its physical content can be assessed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes replacing the orbital-rotation optimizer in natural orbital functional (NOF) calculations with an ADAM/AMSGrad-style adaptive moment scheme, alternated with softmax-parameterized conjugate-gradient optimization of occupation numbers. The authors apply the resulting DoNOF implementation to three large systems: a 10x10x10 hydrogen cube (1000 electrons) with PNOF7, fullerenes C36/C60 with PNOF7/GNOF, and linear acenes with PNOF7, GNOF, and a modified GNOF (GNOFm). The central claims are that the optimizer dramatically increases the accessible scale of NOF calculations and that the hydrogen cube exhibits a metal-to-insulator transition with a critical distance rc of about 2.46 Å.","tokens_in":13780,"tokens_out":3371,"duration_ms":32295,"significance":"If the convergence and scalability claims are substantiated, the work would be an important practical advance: NOF calculations on thousands of correlated electrons are rare, and a robust gradient-based orbital rotation optimizer is a natural fit for the field. The paper deserves credit for stating the algorithm in detail, including hyperparameters and the AMSGrad max-second-moment modification, and for reporting a nontrivial 1000-electron PNOF7 calculation. However, the validation is currently qualitative: there is no stationarity-based convergence criterion for the large run, no comparison with previous NOF optimizers in cost or iteration counts, and the acene agreement relies on a post hoc functional modification. These gaps leave the central claims plausible but not yet fully established.","major_comments":[{"comment":"The stopping rule for the outer loop is based on failure to improve the energy and on iteration budgets, not on a gradient-norm or energy-change threshold, so the 1000-electron PNOF7 curves in Fig. 1 could terminate at non-stationary points. Because Eq. (11) uses a monotonically nondecreasing max second moment, step sizes shrink throughout the run and a stalled slow crawl can masquerade as convergence; the paper should report convergence diagnostics for the 1000-electron run (e.g., final gradient norm, energy change per external iteration, restart or reinitialization checks) and, for at least one smaller hydrogen cluster, verify stationarity against a trusted optimizer or direct gradient threshold.","section":"SI Algorithm 1; Eq. (11)"},{"comment":"GNOFm is constructed after GNOF is found to be offset from the experimental acene gaps, and the same experimental data are then used to claim agreement; this makes the acene validation circular for the GNOFm points. The modified static interpair term (the new first sum in Eq. (32) plus the prefactor change from 1/2 to 1) needs independent physical justification and testing on systems not used in its construction; otherwise the GNOFm results in Fig. 3 should be presented as a fit rather than a prediction.","section":"Acenes, Fig. 3; SI GNOFm, Eq. (32)"},{"comment":"The quantitative content of the metal-to-insulator transition, specifically rc approximately 2.46 Å, rests on PNOF7 being accurate at the 1000-electron scale, yet the functional is validated only on smaller clusters (Ref. 47) and no systematically improvable reference is given for the cube. A benchmark at intermediate cluster sizes against FN-DMC or another size-consistent reference would establish that size-dependent functional error does not shift the transition; without it, the largest-calculation claim is a demonstration of optimizer capability rather than of physical accuracy at scale.","section":"Fig. 1; Ref. 47"},{"comment":"The paper's practical-advance claim is not quantified: no wall-clock timings, iteration counts, or comparisons with the existing iterative-diagonalization NOF optimizer are reported for any of the three applications. Adding such data would show that the improvement is due to the optimizer rather than to hardware or implementation details; at present the reader cannot assess the magnitude of the claimed 'substantial advance' in computational feasibility.","section":"Introduction and Applications (no timing data)"}],"minor_comments":[{"comment":"There are typos that should be corrected: 'algortithm' in the algorithm caption, 'convergency' near Eq. (18), and 'interpar' instead of 'interpair' in the GNOFm section.","section":"SI Algorithm 1 title and text"},{"comment":"The sentence 'These similarities allow us to drawn inspiration' is ungrammatical and should read 'allow us to draw inspiration'.","section":"Full text, paragraph after Eq. (11)"},{"comment":"Reference numbering is inconsistent: the label [4] is used for both Levy and for Franco et al., and the label [3] is used for both Valone and Kingma and Ba; the bibliography should be renumbered and all in-text citations checked.","section":"References"},{"comment":"The GNOFm derivation refers to terms 'marked in blue', but the color marking is not visible in monochrome print or for color-blind readers; use explicit labels such as (new) or (modified) instead.","section":"SI GNOFm section"},{"comment":"The basis-set convergence analysis is performed on the H8 cube, not on the 1000-electron cube; a sentence should explicitly state that this is a proxy analysis and that basis-set convergence at the 1000-electron scale has not been directly demonstrated.","section":"SI Basis Set Convergence Analysis"}],"recommendation":"major_revision","confidential_remarks":"This is a potentially valuable methods paper from an established group, and the 1000-electron PNOF7 calculation is an impressive demonstration. The main risks are the absence of convergence diagnostics for the flagship calculation and the post hoc construction of GNOFm followed by comparison with the same acene data used to motivate it. I would encourage the editor to require stationarity checks or a benchmark against a trusted optimizer, plus a non-circular test of GNOFm, before publication. The fit with the journal's scope is good."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead the Lew-Yee/del Campo/Piris NOF paper. The genuinely new thing is the orbital-rotation optimizer: ADAM with a max-based second moment (AMSgrad-style), with the step taken from y=0 each time, alternated with softmax-parameterized occupation-number optimization. That combination is not in the prior NOF literature, and the algorithmic detail in the SI is solid. The 1000-electron hydrogen cube PNOF7 curve is a real computational milestone—I think it is the largest NOF calculation to date—and the fullerene occupation plots are a nice sanity check. The paper is worth a serious referee.\n\nWhere it gets soft, in order of severity. First, the stress-test concern about convergence diagnostics is legitimate. Algorithm 1 in the SI stops or shrinks the learning rate when the energy does not improve; it never thresholds on the gradient norm or energy change. So the 1000-electron curve could be a slow crawl along a flat valley rather than a converged minimum. The smoothness of the dissociation curve does not fix this—flat directions are exactly what you get at stretched H–H distances. No iteration counts, gradient norms, or restart checks are reported. That undermines the quantitative claim about the critical distance rc, and even the \"largest NOF calculation\" claim needs a statement of what \"converged\" means. Second, GNOFm is post-hoc: they modify the functional after seeing GNOF's offset from experimental acene gaps, then use it to reproduce those same gaps. They do flag this as preliminary, but the figure and text read as validation, not fitting. Third, there is no benchmark against a systematically improvable method for the 1000-electron system, so the PNOF7 curve's physical content is unvalidated at that scale.\n\nMinor: the promotional language (\"disruptive,\" \"complete change in size scale\") is over the top, and there is no timing comparison against their own older optimizer or anyone else's, so the computational advantage is asserted rather than demonstrated.\n\nThe core contribution—the optimizer itself—is likely sound and useful to the NOF community. But as written, the strong scaling claims outrun the evidence. I would not accept this as-is, but with iteration counts, a stationarity measure, and an honest reframing of GNOFm as a fitted modification, it could be a solid methods paper.","headline":"A potentially useful ADAM-based optimizer for natural orbital functional calculations, with a flagship 1000-electron run, but the convergence claim needs hardening and the GNOFm acene fit is circular.","tokens_in":14279,"tokens_out":1018,"would_cite":true,"duration_ms":12205,"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":"Deep-learning-inspired optimizer takes natural orbital functional calculations to 1000 correlated electrons.","keywords":["natural orbital functional theory","strongly correlated electrons","ADAM optimizer","orbital rotation","occupation numbers","metal-insulator transition","PNOF7","GNOF"],"falsifier":"Compute the 10×10×10 H-cube dissociation curve with an independent, systematically improvable method such as fixed-node diffusion Monte Carlo and compare the critical distance at which the off-diagonal 1RDM measure γ falls below 0.05; if the NOF curve places the transition at a materially different r_c or energy, the claim that the method captures the metal-to-insulator transition at this scale fails.","tokens_in":13251,"feed_emoji":"⚛️","tokens_out":9522,"duration_ms":74641,"temperature":0.7,"pith_summary":"Natural orbital functional (NOF) theory can in principle describe strongly correlated electrons at low cost, but its orbital-optimization step converges slowly, limiting practical use to small systems. This paper claims that importing the ADAM optimizer from deep learning, with a max-based second moment, learning-rate annealing, and a softmax parameterization of occupation numbers, removes that bottleneck. The authors demonstrate the scheme on the symmetric dissociation of a 10×10×10 hydrogen cube containing 1000 electrons with the PNOF7 functional, the largest NOF calculation reported to date, and on fullerenes and linear acenes with PNOF7 and GNOF. If correct, the method makes NOF theory a practical tool for thousands of correlated electrons and for studying physics such as the metal-to-insulator transition in a homogeneous electron system.","feed_headline":"Deep-learning optimizer reaches 1000 strongly correlated electrons","feed_subtitle":"An ADAM-based orbital update clears the convergence bottleneck that kept natural orbital functional theory below a few hundred electrons.","key_machinery":"The central object is the ADAM-with-max-second-moment optimizer applied to orbital rotations. At each step the orbital gradient $g_{pq}=4(\\lambda_{pq}-\\lambda_{qp})$ for $p<q$ drives first and second moments $m_t$ and $v_t$; the second moment is replaced by its historical maximum $\\hat{v}^{\\max}_t = \\max(\\hat{v}^{\\max}_{t-1}, \\hat{v}_t)$, and the rotation step is $y_t = \\alpha \\hat{m}_t / \\sqrt{\\hat{v}^{\\max}_t + \\epsilon}$ with $U_t = e^{y_t}$ and $C_t = C_{t-1}U_t$. Because derivatives are always evaluated at $y=0$, each step starts from the origin and transfers gradient history through the moments; the maximum operation is essential for convergence. Occupations are optimized separately with the conjugate-gradient method under the softmax parameterization, which builds in the N-representability bounds.","core_discovery":"The paper's central claim is that the convergence bottleneck in NOF calculations is the natural-orbital rotation step, and that adaptive-moment optimization—specifically ADAM with the historical maximum of the second moment (AMSgrad-style)—converges these rotations reliably without Hessian information, while occupations are optimized separately under a softmax parameterization that enforces N-representability. Alternating these two cycles with early stopping and a scheduled learning rate, the authors converge PNOF7 on a 1000-electron hydrogen cube and reproduce the expected metal-to-insulator transition: the harmonic average of off-diagonal 1RDM elements in the atomic-orbital basis drops to near zero at the expected critical distance. They further show occupancy distributions for C36 and C60 and singlet-triplet gaps across the n-acene series, introducing a modified GNOFm functional that improves agreement with experimental gaps.","pith_inferences":["The ADAM-based rotation optimizer is likely transferable to other one-particle reduced density matrix functional theories and to natural-orbital optimization in multiconfigurational self-consistent field, where the same y=0 gradient bottleneck appears.","The convergence schedule may be further accelerated by mini-batching or preconditioning of the orbital gradients, techniques common in deep learning but untested here.","If PNOF7's size-dependent accuracy holds, the hydrogen-cube curve could serve as a reference for functional development and for benchmarking approximate methods that describe Mott physics.","A direct test would be to apply the same optimizer to a periodic or embedded version of NOF theory, extending the method to solids and surfaces."],"forward_implications":["NOF calculations become practical for systems with thousands of correlated electrons, at least in minimal and polarized double-zeta basis sets.","The 10×10×10 hydrogen cube provides a benchmark strong-correlation problem where all 1000 electrons participate in the correlation phenomenon.","The method captures a metal-to-insulator transition without symmetry breaking or system-specific tuning of the functional.","Fullerenes with hundreds of spatial orbitals can be probed for static correlation via fractional occupation numbers.","Singlet-triplet gaps in long acenes can be estimated with the modified GNOFm functional, which improves agreement with experimental gaps."],"supporting_citations":[{"why":"Supplies the ADAM optimizer whose moment updates and bias correction form the core of the new orbital-rotation scheme.","marker":"[3]"},{"why":"Defines the PNOF7 functional used for the 1000-electron hydrogen cube and the fullerene calculations.","marker":"[16]"},{"why":"Defines the GNOF functional used for fullerenes, acenes, and the modified GNOFm variant.","marker":"[22]"},{"why":"Demonstrates PNOF7 reproduces dissociation curves of smaller hydrogen clusters, the basis for trusting the 1000-electron curve.","marker":"[47]"},{"why":"Provides the DoNOF software in which the optimization scheme and all calculations are implemented.","marker":"[41]"},{"why":"Establishes the iterative diagonalization method, the previous state of the art that the ADAM-based scheme replaces to remove the convergence bottleneck.","marker":"[36]"}],"fun_headline_variants":["ADAM-optimized NOF reaches 1000 correlated electrons","Deep learning reignites NOF theory to 1000 electrons","Adaptive momentum accelerates NOF to 1000 electrons","NOF hits 1000 electrons with deep learning optimizer","Machine learning scales NOF to record 1000 electrons"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole demonstration rests on PNOF7 being as reliable for a 1000-electron cluster as it is for the small clusters it was tested on, and the paper provides no systematically improvable reference calculation for the large dissociation curve.","fun_headline_variants_meta":{"raw":{"variants":["ADAM-optimized NOF reaches 1000 correlated electrons","Deep learning reignites NOF theory to 1000 electrons","Adaptive momentum accelerates NOF to 1000 electrons","NOF hits 1000 electrons with deep learning optimizer","Machine learning scales NOF to record 1000 electrons"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000452,"raw_usage":{"total_tokens":2311,"prompt_tokens":1016,"completion_tokens":1295,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":632,"completion_tokens_details":{"reasoning_tokens":1211}},"tokens_in":632,"tokens_out":1295,"duration_ms":10098,"temperature":1.0,"reasoning_tokens":1211,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:08:19.014450+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the 10×10×10 H-cube dissociation curve with an independent, systematically improvable method such as fixed-node diffusion Monte Carlo and compare the critical distance at which the off-diagonal 1RDM measure γ falls below 0.05; if the NOF curve places the transition at a materially different r_c or energy, the claim that the method captures the metal-to-insulator transition at this scale fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the ADAM optimizer whose moment updates and bias correction form the core of the new orbital-rotation scheme."},{"cited_title":"Piris, X","cited_arxiv_id":null,"evidence_quote":"Defines the PNOF7 functional used for the 1000-electron hydrogen cube and the fullerene calculations."},{"cited_title":"Piris, Phys","cited_arxiv_id":null,"evidence_quote":"Defines the GNOF functional used for fullerenes, acenes, and the modified GNOFm variant."},{"cited_title":"Hongo, M","cited_arxiv_id":null,"evidence_quote":"Demonstrates PNOF7 reproduces dissociation curves of smaller hydrogen clusters, the basis for trusting the 1000-electron curve."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the DoNOF software in which the optimization scheme and all calculations are implemented."},{"cited_title":"Szabo and N","cited_arxiv_id":null,"evidence_quote":"Establishes the iterative diagonalization method, the previous state of the art that the ADAM-based scheme replaces to remove the convergence bottleneck."}],"review_version":1}