Advancing Natural Orbital Functional Calculations Through Deep Learning-Inspired Techniques for Large-Scale Strongly Correlated Electron Systems

Jorge M. del Campo; Juan Felipe Huan Lew-Yee; Mario Piris

REVIEW 4 major objections 5 minor 1 cited by

Advancing Natural Orbital Functional Calculations Through Deep Learning-Inspired Techniques for Large-Scale Strongly Correlated Electron Systems

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Deep-learning-inspired optimizer takes natural orbital functional calculations to 1000 correlated electrons.

desk verdict 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. read the letter →

arxiv 2411.18493 v2 pith:AT6DEZ3H submitted 2024-11-27 cond-mat.str-el physics.chem-phphysics.comp-ph

Juan Felipe Huan Lew-Yee , Jorge M. del Campo , Mario Piris This is my paper · ORCID

classification cond-mat.str-elphysics.chem-phphysics.comp-ph

keywords naturalorbitalfunctionaltheorystronglycorrelatedelectronsADAMoptimizerrotationoccupationnumbersmetal-insulatortransitionPNOF7GNOF

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Read 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.

Where 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.

Minor: 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.

The 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.

Referee Report

4 major / 5 minor

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.

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 (4)

[SI Algorithm 1; Eq. (11)] 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.
[Acenes, Fig. 3; SI GNOFm, Eq. (32)] 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.
[Fig. 1; Ref. 47] 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.
[Introduction and Applications (no timing data)] 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.

minor comments (5)

[SI Algorithm 1 title and text] 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.
[Full text, paragraph after Eq. (11)] The sentence 'These similarities allow us to drawn inspiration' is ungrammatical and should read 'allow us to draw inspiration'.
[References] 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.
[SI GNOFm section] 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.
[SI Basis Set Convergence Analysis] 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.

Circularity Check

1 steps flagged · score 3.0 of 10

Minor circularity in the GNOFm acene-gap claim; the ADAM scaling results are not circular.

fitted input called prediction [Main text, linear acenes paragraph (Fig. 3) and Supplemental Material 'GNOF modified (GNOFm)']
"While GNOF produces a qualitatively correct curve shape, it appears to be offset relative to the experimental values. In this work, we have modified the GNOF formulation (see Supplemental Material) to include interactions between strongly occupied orbitals in the antiparallel spin blocks, resulting in what we call 'modified GNOF' (GNOFm), represented by the green marks. Interestingly, these terms appear to be crucial for predictions that align more closely with the experimental data."

The acene singlet-triplet gaps are presented as GNOFm 'predictions' in Fig. 3, but GNOFm is introduced in this same paper because GNOF was 'offset relative to the experimental values.' The modification, which adds static interpair interactions between occupied orbitals and changes a constant from 1/2 to 1 (SI Eq. 32), is chosen after seeing the experimental acene gaps. Therefore the closer agreement between GNOFm and experiment for these acenes is an in-sample consequence of that adjustment rather than an independent prediction. The paper itself flags this as requiring further investigation, and the central ADAM optimization claim is unaffected because it is tested on systems independent of the optimizer's construction.

full rationale

The paper's core contribution is a deep-learning-inspired optimization scheme (ADAM with AMSGrad-style max second moment and softmax ON parameterization) for NOF calculations. That scheme is assessed on a 1000-electron hydrogen cube, fullerenes, and acenes, which are independent of the optimizer design; no parameter of the optimizer is fitted to those results, so the scaling and convergence claims are not circular. The main circularity concern is confined to the acene application: GNOFm is modified after observing GNOF's offset from experimental singlet-triplet gaps and then used to 'predict' those same gaps, making the improved agreement a post-hoc fit. This is a real but localized circular step. The self-citations to prior PNOF7 and GNOF developments are normal scientific references and are not load-bearing for the optimizer claim, since those functionals are used as tools and the optimization is validated on new systems. The absence of a gradient-norm convergence threshold in Algorithm 1 is a correctness and reproducibility risk, not a circularity.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central method depends on default choices for optimizer hyperparameters and on the transferability of previously published NOF functionals. GNOFm is a new functional modification with no independent derivation. No new physical entities are introduced.

free parameters (6)

Learning rate initial alpha = 0.01
Set by hand; scheduled reduction by factor p=0.4 when energy does not improve (Algorithm 1).
Momentum hyperparameters beta1, beta2 = beta1=0.7, beta2=0.9
Chosen instead of deep learning defaults (0.9, 0.999) to improve convergence; no systematic scan reported.
Learning rate decay factor p = 0.4
Annealing factor applied to alpha when no energy improvement (Algorithm 1).
Initial orbital iterations N_orb_it = 10
Early-stopping schedule: increased by 10 when convergence stalls (Algorithm 1).
Gamma transition threshold = 0.05
Used to declare off-diagonal 1RDM elements 'effectively close to zero' at the transition; arbitrary.
GNOFm prefactor for static interpair term = 1
Changed from 1/2 in GNOF to 1; appears to be adjusted to improve acene gap agreement (Eq. 32).

assumptions (5)

domain assumption PNOF7/GNOF accurately describe electron correlation in the studied systems
The paper treats PNOF7 and GNOF as accurate enough to interpret the H cube transition and acene gaps; prior validation exists only for smaller systems (Ref. 47).
domain assumption The harmonic average gamma of off-diagonal 1RDM elements in the AO basis is a valid order parameter for the metal-insulator transition
Equation (12) uses gamma to identify rc; the criterion |Gamma| < 0.05 is arbitrary.
domain assumption The softmax parameterization of occupation numbers is stable and preserves N-representability
Used to make ON optimization unconstrained; no proof of global convergence.
domain assumption STO-3G basis is sufficient for the 1000-electron H cube dissociation curve
Basis-set convergence is only demonstrated for the H8 cube, not the 1000-electron system.
ad hoc to paper GNOFm modification is physically justified and does not bias the acene predictions
The modification is introduced after observing GNOF's offset from experimental acene gaps; no independent derivation.

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Cite this review

Pith. "Pith review of Advancing Natural Orbital Functional Calculations Through Deep Learning-Inspired Techniques for Large-Scale Strongly Correlated Electron Systems." pith.science (2026). https://pith.science/paper/AT6DEZ3H

@misc{pith2026241118493,
  author       = {Pith},
  title        = {Pith review of: Advancing Natural Orbital Functional Calculations Through Deep Learning-Inspired Techniques for Large-Scale Strongly Correlated Electron Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AT6DEZ3H}},
  note         = {Machine review of arXiv:2411.18493}
}

read the original abstract

Natural orbital functional (NOF) theory offers a promising approach for studying strongly correlated systems at an affordable computational cost, with an accuracy comparable to highly demanding wavefunction-based methods. However, its widespread adoption in cases involving a large number of correlated electrons has been limited by the extensive iterations required for convergence. In this work, we present a disruptive approach that embeds the techniques used for optimization in deep learning within the NOF calculation, constituting a substantial advance in the scale of accessible systems. The revamped procedure is based on the adaptive momentum technique for orbital optimization, alternated with the optimization of the occupation numbers, significantly improving the computational feasibility of challenging calculations. This work represents a complete change in the size scale of the systems that can be reached using NOF theory. We demonstrate this with three examples that involve a large number of electrons: (i) the symmetric dissociation of a large hydrogen cluster, (ii) an analysis of occupancies distribution in fullerenes, and (iii) a study of the singlet-triplet energy gap in linear acenes. Notably, the hydrogen cluster calculation, featuring 1000 electrons, represents the largest NOF calculation performed to date and one of the largest strongly correlated electron calculations ever reported. This system, which serves as an ideal model for a strongly correlated Mott insulator, illustrates a metal-to-insulator transition where all electrons participate in the correlation phenomenon, offering insight in a unique challenge. We anticipate that this work will enable the practical application of NOFs to increasingly complex and intriguing systems, leveraging the method's inherent scalability and accuracy.

Figures

Figures reproduced from arXiv: 2411.18493 by the authors.

**Figure 3.** FIG. 3. Singlet-triplet energy gap in the linear n-acene series. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** FIG. 4. Evolution of the internal iterations of natural orbitals [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: FIG. 5. Total energy of the H [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: shows the dissociation curves for the H8 cube computed using the cc-pVXZ and aug-cc-pVXZ basis sets for X={D,T,Q}. The overall agreement between the different basis sets demonstrates the stability of the NO optimization scheme and confirms that the dissociation trends…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Enhancing the Computational Efficiency of the DoNOF Program through a New Orbital Sorting Scheme
physics.chem-ph 2025-02 conditional novelty 6.0 of 10

An alternating orbital sorting scheme enables warm starts in DoNOF calculations, and a two-step (perfect pairing to full subspace) strategy reduces computational cost while maintaining accuracy.

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