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Optimizing adsorption configurations on alloy surfaces using Tensor Train Optimizer

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that truncating the adsorption-energy expansion at third order is sufficient to approximate adsorption energies within 0.043 eV per molecule and to find the most stable CO/NO configurations on binary alloy slabs and…

desk verdict Solid internal demonstration that third-order HUBO beats QUBO for adsorption search, but the 'chemical accuracy' claim is only against the MLIP that built it. read the letter →

arxiv 2507.20827 v1 pith:TEG6WFUN submitted 2025-07-28 physics.chem-ph

Tuan Minh Do , Tomoya Shiota , Wataru Mizukami This is my paper · ORCID
classification physics.chem-ph
keywords adsorptionconfigurationHUBOtensortrainTTOptmachinelearninginteratomicpotentialbinaryalloyhigh-entropyglobaloptimization
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

This paper tries to establish that the exponentially hard search for the most stable arrangement of adsorbed molecules on alloy surfaces can be solved by writing the adsorption energy as a sum of one-, two-, and three-body site interactions and optimizing that expression with the Tensor Train Optimizer (TTOpt), a classical method that needs no specialized hardware. The central claim is that stopping at third order suffices: for CO and NO on 13 binary alloy (111) slabs and on 25 five-element high-entropy alloy nanoparticles, the third-order (HUBO) approximation reproduces the reference machine-learned potential within 0.043 eV per molecule across nearly the entire coverage range, and it identifies configurations that are at least as stable as those found by a genetic algorithm. This matters because quantum and digital annealers, the current alternatives, are limited to quadratic cost functions and require specialized hardware. The paper also reports that HUBO helps TTOpt itself: third-order terms preserve correlations between adsorption sites, so the optimizer finds lower-energy configurations than it does with the quadratic-only QUBO form.

What carries the argument

The load-bearing machinery is the HUBO expansion of the adsorption energy (equation 10 of the paper) combined with TTOpt, a classical optimizer that treats the energy as an implicitly defined tensor with one binary dimension per adsorption site and searches for the lowest entry using a low-rank tensor-train representation. TTOpt builds the tensor train through TT-cross approximation with maxvol submatrix selection, using the mapping $E'(\sigma)= \pi/2 - \arctan(E(\sigma)-E_{\min})$ so that the maxvol algorithm can rank entries regardless of sign; a coverage penalty $E_{\mathrm{penalty}} = E_0 (n_{\mathrm{target}} - n_{\mathrm{trial}})^2$ with $E_0=10\,\mathrm{eV}$ constrains the search to a fixed number of adsorbates. The third-order terms are the part of the model that does the real work: they capture the three-body repulsion that sets in above roughly 0.4–0.5 coverage and that makes the quadratic QUBO landscape misleading.

What would settle it

Take the optimal configurations that TTOpt finds with the HUBO cost function on, say, CO/Pd-Pt(111) and on one IrPdPtRhRu nanoparticle, and recompute their adsorption energies with a density-functional method (or with a different high-accuracy MLIP) at coverages 0.5, 0.75, and 1.0; if the mean absolute deviation between MACE-Osaka24 and the reference exceeds roughly 0.043 eV per molecule, then the chemical-accuracy conclusion does not extend to physical adsorption energies, although the internal comparison between QUBO and HUBO truncations would remain valid.

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Extended reading notes

Core claim

The paper's central discovery is that adsorption of CO and NO on alloy surfaces can be cast as a higher-order unconstrained binary optimization (HUBO) problem, with a cost function $E_{\mathrm{HUBO}} = \sum_{i} U_{iii} \sigma_i + 6 \sum_{i<j} U_{ijj} \sigma_i \sigma_j + 6 \sum_{i<j<k} U_{ijk} \sigma_i \sigma_j \sigma_k$, and that the coefficients $U_{iii}$, $U_{ijj}$, and $U_{ijk}$, fixed by all uni-, bi-, and trimolecular adsorption energies computed with the MACE-Osaka24 potential, make the HUBO expression accurate enough to identify the global minimum configuration. On the Pd-Pt(111) slab, the quadratic-only QUBO version deviates from the reference energies once coverage exceeds 0.5 CO per surface atom and even reverses the correct trend above 0.75, while the HUBO version stays within 0.043 eV per molecule up to saturation. Benchmarks across 13 binary alloys for both CO and NO show the same pattern, and on high-entropy alloy nanoparticles the HUBO error is close to zero across the full coverage range. TTOpt with the HUBO cost function also outperforms a genetic algorithm at high coverage on nanoparticles, which the authors attribute to third-order terms preserving the correlations between adsorption sites that the optimizer can exploit.

Load-bearing premise

The reference 'exact' adsorption energies used to build and test the HUBO model come from the MACE-Osaka24 machine-learned potential rather than from DFT or experiment, so the reported 'chemical accuracy' claim transfers to physical adsorption energies only if that potential is itself accurate to within about 0.043 eV for these alloy and adsorbate systems.

Editorial extensions

If this is right

  • On the 13 binary alloy slabs, the quadratic QUBO formulation stays within 0.043 eV per molecule only up to about 0.4–0.6 coverage, while including third-order terms extends this range to near saturation for more than half of the alloys.
  • For the high-entropy alloy nanoparticles, the HUBO error is close to zero across the whole 0–100% coverage range, and the optimal CO configurations differ from the QUBO ones once coverage exceeds 0.25 per surface atom.
  • TTOpt with the HUBO cost function consistently beats a heavily tuned genetic algorithm at high CO coverages on the nanoparticles, whereas with the QUBO cost function the genetic algorithm wins at moderate coverages because the quadratic approximation distorts the energy landscape.
  • The full workflow runs on ordinary CPUs with no specialized hardware, and the practical bottleneck shifts to constructing the HUBO tensor: roughly 250 GPU-hours for one nanoparticle versus 3 hours for the QUBO matrix.

Reading between the lines

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

  • If third-order truncation is as accurate as the paper finds, then a distance-based cutoff for the three-body tensor should reproduce the full HUBO results; the paper's own distance data (CO–CO separations of 3.7 Å at 0.5 coverage and 2.8 Å at 0.75 coverage on Pd-Pt) suggest a concrete way to cut the 250-hour tensor construction cost by orders of magnitude.
  • The sufficiency of third order is probably not special to CO and NO: it should be tested on other adsorbates (O, OH, NHx) to see whether fourth-order terms become necessary when adsorbates bind more strongly, dissociate, or form hydrogen-bond-like networks.
  • Because all energies in the paper trace back to a single machine-learned potential, benchmarking the identified optimal configurations against DFT, or using an uncertainty-aware potential that flags unreliable predictions, would convert the methodological claim into a physical statement about real alloy 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.

Referee Report

3 major / 5 minor

Summary. The paper proposes a workflow for finding the lowest-energy adsorption configurations of CO and NO on alloy surfaces by expressing the adsorption energy as a QUBO or HUBO cost function whose coefficients are obtained from MACE-Osaka24 energies, and then solving the optimization with the tensor-train-based TTOpt method. The authors test the approach on 13 binary alloy (111) slabs and on 25 IrPdPtRhRu high-entropy-alloy nanoparticles, comparing second-order (QUBO) and third-order (HUBO) truncations of the many-body expansion, and benchmarking TTOpt against a genetic algorithm. They report that the HUBO formulation keeps per-adsorbate errors below 0.043 eV relative to MACE-Osaka24 up to high coverages, and that TTOpt with HUBO finds lower-energy configurations than with QUBO, especially at high coverages.

Significance. If the accuracy claim is taken at face value, the paper offers a practical, hardware-free route to higher-order binary optimization for surface-chemistry problems, extending earlier work that was limited to quadratic cost functions and specialized annealers. The study has notable strengths: the coefficient construction is internally consistent (I checked eqs. (4), (8), and (10)-(13) under a symmetric-tensor convention), the benchmark set is broad, the QUBO-versus-HUBO comparison is systematic, the comparison with a genetic algorithm is informative, and the appendix on site-index ordering is a useful practical contribution. However, the headline 'chemical accuracy' claim is measured against MACE-Osaka24, the same model used to generate the HUBO coefficients; it is therefore a truncation-error analysis of an MLIP expansion rather than a validation against physical adsorption energies. The optimality claim is also stronger than what the evidence supports, since neither TTOpt nor GA is an exhaustive solver and no exact enumeration is provided.

major comments (3)
  1. [II.B.1, III (Figs. 4-7)] The 'chemical accuracy' claim is self-referential. All HUBO/QUBO coefficients are derived from MACE-Osaka24 energies (Section II.B.1: 'The energies ... were computed using the MACE neural network potential'), and every 'exact' reference energy used to compute the reported 0.043 eV errors is also a MACE-Osaka24 energy. The errors are therefore truncation errors of a many-body expansion of MACE-Osaka24, not errors with respect to DFT or experiment. Unless MACE-Osaka24 is itself accurate to better than about 0.043 eV for these alloy/adsorbate systems, the abstract's 'within chemical accuracy' claim does not transfer to physical adsorption energies. Please add DFT or experimental benchmarks for at least a representative subset of alloys, coverages, and adsorbates, or revise the abstract and conclusions to state explicitly that the errors are relative to MACE-Osaka24.
  2. [II.A.2, II.B.1 (eqs. (12)-(13))] The treatment of skipped and non-converged configurations is an arbitrary modeling choice that enters the cost function coefficients. The text states that when a configuration has inter-adsorbate distances below 2.3 Å, is a permutation duplicate, or fails to converge in 100 steps, its adsorption energy is set to 10 eV. These 10 eV entries directly affect U_ij and U_ijk via eqs. (12)-(13), and therefore shape the optimization landscape even for configurations that are not themselves skipped. The paper does not report how many configurations were skipped per system, nor how sensitive the identified optima are to the 10 eV substitution. Please quantify the skipped fraction and test the sensitivity of the optimal configurations and energies to the substitution value (e.g., 5 eV, 10 eV, 20 eV) and to the convergence criterion.
  3. [III.D, V] The claim that TTOpt 'identifies optimal adsorption configurations' is not supported by an exhaustive search. TTOpt with maximum TT-rank up to 50 and 10^7 evaluations is a heuristic, and the genetic algorithm comparison in Section III.D shows only relative performance on the truncated QUBO/HUBO objectives, not proximity to the true global minimum. For the small-n cases n=1,2,3 the expansion is exact by construction, but no exhaustive enumeration is provided for any nontrivial n, not even on a small slab where the configuration space would be manageable. Please either add exhaustive validation for a small system (e.g., n=4 or n=5 on a reduced site set) or reframe the abstract and conclusion to say that TTOpt finds low-energy configurations rather than provably optimal ones.
minor comments (5)
  1. [II.A.2, eq. (10)] The factor 6 in eq. (10) and the notation U_ijj are confusing unless the reader assumes a fully symmetric tensor convention. Please state explicitly that U is symmetric under permutation of indices and that the reduced form collects all ordered triples with the same multiset of indices.
  2. [II.B.1, Figs. 4-7] The coverage label 'CO per surface atom' is used without defining the number of surface atoms or the mapping from ntarget in eq. (14) to coverage. For the 4x4x4 slab the top layer has 16 atoms and 96 adsorption sites, so 1 CO per surface atom corresponds to n=16; please state this explicitly, as the site count and coverage definition are central to interpreting the figures.
  3. [III.A, Fig. 4c] The error definition is described in the text as the difference between the QUBO/HUBO energy and the MACE-Osaka24 energy for the same configuration, while the figures label it as error per CO or per NO. Please make the denominator explicit in the figure captions and in the main text, and clarify whether the 0.043 eV threshold applies to per-adsorbate or total adsorption energy.
  4. [II.A.2] There is a typo in the sentence 'eqns (7) and (8) were employed to construct the QUBO coefficent matrix' - 'coefficent' should be 'coefficient'.
  5. [IV] The 'chemical accuracy threshold of 0.043 eV' is used throughout without a citation or justification. Since 1 kcal/mol = 0.043 eV is a common convention, please cite a source or state the convention explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the HUBO/QUBO truncation errors are measured against the same MACE-Osaka24 model used to build the coefficients, but the high-coverage tests are genuine third-order truncation checks, not forced fits; the main caveat is that 'chemical accuracy' is MACE-relative and depends on a self-cited model's external accuracy.

full rationale

The paper's derivation chain is internally non-circular. HUBO coefficients (Eqs. 11-13) are constructed from MACE-Osaka24 uni-, bi-, and trimolecular adsorption energies, and make the HUBO expression exact for n <= 3 by construction. For the reported high-coverage cases (up to 96 adsorbates on slabs and 170-site nanoparticles), the HUBO is a genuine third-order truncation of the many-body expansion, and the error metric -- the difference between the truncated QUBO/HUBO energy and the MACE-Osaka24 energy for the same configuration (Section III.A, Fig. 4c) -- tests whether higher-order correlations are negligible. This is falsifiable and empirically nontrivial: QUBO errors grow sharply above 0.5 coverage and even reverse the energy trend, while HUBO errors remain below 0.043 eV, showing the benchmark is not vacuous. The TTOpt-versus-GA comparison (Section III.D) is also independent and not circular, and TTOpt does not uniformly win, further indicating a meaningful optimization benchmark. The only concern is semantic and evidential, not a circular reduction: the paper calls eqn (1) with MACE-Osaka24 energies the 'exact expression' and labels MACE-relative truncation errors as 'chemical accuracy' (0.043 eV), but MACE-Osaka24 is the authors' own in-house trained potential (ref. 11, Shiota et al., with overlapping authorship), and no DFT or experimental benchmark for CO/NO on the 13 binary alloys or HEA nanoparticles is provided. Thus the physical transfer of the 'chemical accuracy' claim rests on the external accuracy of a self-cited model. That is a validation gap and a correctness risk, not a circular derivation: the high-coverage MACE energies used for the error analysis are not inputs into the HUBO coefficient construction. The mathematical result that third-order terms suffice to reproduce MACE-Osaka24 stands on its own.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central accuracy claim rests on the MACE-Osaka24 model, whose weights are fitted elsewhere, on the truncation at third order, on the discrete ACAT site model, and on two hand-set 10 eV energy values. These are the main assumptions the reader is asked to accept; the paper provides no external DFT or experimental benchmark to validate them.

free parameters (3)
  • E_penalty0 = 10 eV
    Prefactor of the coverage constraint in eq (14). Hand-set; too small fails to enforce n_target, too large may dominate the landscape.
  • E_skipped = 10 eV
    Energy assigned to any 1-, 2-, or 3-adsorbate configuration skipped for short distances, permutation equivalence, or non-convergence (Section II.B.1). This arbitrary value enters the coefficient tensor.
  • MACE-Osaka24 weights = Pre-trained 'large' model
    The reference energy model is itself a fitted MLIP. All accuracy claims inherit its systematic error, which is not quantified against DFT here.
assumptions (5)
  • domain assumption MACE-Osaka24 is an accurate reference for adsorption energies on the tested alloys
    Invoked in Section II.B.1 where all Etotal, Esubstrate, Eadsorbate are computed with MACE, and errors are measured against it. No DFT or experimental benchmark is provided.
  • domain assumption Fourth- and higher-order interaction terms do not affect optimal configurations or energies for these systems
    The HUBO cost function stops at third order; the "sufficiency" conclusion is tested only indirectly on TTOpt-selected configurations.
  • domain assumption Discrete occupation of the 96 or 170 ACAT sites captures all relevant adsorption physics
    Adsorbates are restricted to predefined sites with fixed perpendicular relaxation; adsorbate tilt, surface reconstruction, and subsurface sites are excluded.
  • ad hoc to paper Skipped or non-converged configurations are faithfully replaced by E=10 eV
    Section II.B.1 assigns 10 eV to skipped configurations; this is an unvalidated approximation that feeds into the HUBO coefficients.
  • domain assumption TTOpt with rank scan 2-50 and 1e7 evaluations finds the global minimum
    For slabs no exhaustive verification is reported; for nanoparticles only GA is used as a cross-check (Section III.D).

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Cite this review

Pith. "Pith review of Optimizing adsorption configurations on alloy surfaces using Tensor Train Optimizer." pith.science (2026). https://pith.science/paper/TEG6WFUN

@misc{pith2026250720827,
  author       = {Pith},
  title        = {Pith review of: Optimizing adsorption configurations on alloy surfaces using Tensor Train Optimizer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TEG6WFUN}},
  note         = {Machine review of arXiv:2507.20827}
}
read the original abstract

Understanding how molecules arrange on surfaces is fundamental to surface chemistry and essential for the rational design of catalytic and functional materials. In particular, the energetically most stable configuration provides valuable insight into adsorption-related processes. However, the search for this configuration is a global optimization problem with exponentially growing complexity as the number of adsorbates and possible adsorption sites increases. To address this, we express the adsorption energy as a sum of multi-adsorbate interaction terms, evaluated using our in-house trained machine learning interatomic potential MACE-Osaka24, and formulate the search for the most stable configuration as a higher-order unconstrained binary optimization (HUBO) problem. We employ a tensor-train-based method, Tensor Train Optimizer (TTOpt), to solve the HUBO problem and identify optimal adsorption configurations of CO and NO molecules on various alloys up to full surface coverage. Our results show that including interaction terms up to third order may be sufficient to approximate adsorption energies within chemical accuracy and to identify optimal configurations. We also observed that TTOpt performs better with the HUBO formulation, suggesting that third-order terms help preserve correlations between adsorption sites, which allow TTOpt to optimize configurations more effectively. The extensive benchmarks across various alloys, surface geometries, and adsorbates demonstrate the robustness and applicability of using TTOpt to solve HUBO-type global optimization problems in surface chemistry. In contrast to quantum and digital annealers, which have recently been applied to similar global optimization tasks but are restricted to cost functions with at most quadratic terms, our approach can incorporate higher-order terms in a straightforward manner and does not require specialized hardware.

Figures

Figures reproduced from arXiv: 2507.20827 by the authors.

Figure 1
Figure 1. provides a schematic overview of the workflow adopted in this work to identify optimal adsorption config￾urations using the HUBO formulation of the adsorption en￾ergy. First, all possible uni-, bi-, and trimolecular adsorption energies were computed and subsequently used to build the HUBO coefficient tensor according to eqns (11)-(13). Using the obtained coefficients Ui jk, the HUBO formulation of the adsorption ene… view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. a shows which adsorption sites correspond to the first three occupation numbers σ1, σ2, and σ3 of the configuration σ. The structure of E along these three dimensions is de￾picted in Fig. 3b. Since each dimension has size 2, there are eight entries in total, each representing a unique adsorption configuration. For illustration, the configurations correspond￾ing to three of the entries are explicitly shown. The value… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: shows the difference in adsorption energy between the configurations identified by the two methods, with nega￾tive values and positive values indicating more stable config￾urations from TTOpt and GA, respectively. Note that since we are evaluating the performance of th…

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