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arxiv: 2605.15927 · v1 · pith:QJ56JVHEnew · submitted 2026-05-15 · ⚛️ physics.chem-ph

Data-driven complete basis set limit estimates from a minimal auxiliary basis

Nicolas Grimblat , Gabriel Klassen , Guido Falk von Rudorff This is my paper

Pith reviewed 2026-05-19 19:03 UTC · model grok-4.3

classification ⚛️ physics.chem-ph
keywords complete basis set limitkernel ridge regressionmachine learningquantum chemistrybasis set incompletenessCABSpairwise interaction modelChebyshev polynomials
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The pith

A pairwise interaction model plus minimal CABS baseline and kernel ridge regression estimates the complete basis set limit from a single minimal-basis calculation.

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

The paper proposes estimating the complete basis set limit energy with only one quantum chemistry run in a minimal basis set. It begins with a physical baseline that combines a pairwise interaction model and a minimal complementary auxiliary basis set to account for the dominant incompleteness effects, then trains kernel ridge regression on molecular representations to correct the residual error. This hybrid route is presented as more efficient than training machine learning models directly on energies or using delta-learning corrections. The regression step is made practical by approximating atom-wise local kernels with Chebyshev polynomials, which reduces the compute needed for training on moderate resources. A reader would care because conventional CBS extrapolations demand multiple calculations at progressively larger basis sets, driving up cost for each new molecule.

Core claim

The central claim is that the CBS energy can be estimated from a single quantum chemistry calculation in a minimal basis set by combining a pairwise interaction model with a minimal complementary auxiliary basis set baseline and applying a kernel ridge regression correction to the remaining error, which is more efficient than both direct and delta-machine learning, with the kernel models made tractable by Chebyshev polynomial approximations to atom-wise local kernels.

What carries the argument

Kernel ridge regression that corrects residuals after a pairwise interaction model augmented by a minimal CABS baseline, with atom-wise local kernels approximated via Chebyshev polynomials.

If this is right

  • CBS energies become available after a single minimal-basis calculation rather than a series of increasing basis-set sizes.
  • The physical baseline reduces the learning burden on the regression model, allowing smaller training sets to reach useful accuracy.
  • Chebyshev approximations to local kernels lower the computational cost of training, making the method feasible on moderate hardware.
  • The same hybrid baseline-plus-correction pattern can be applied to other slowly converging quantities in quantum chemistry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be tested on larger or more diverse molecular datasets to check whether the baseline still leaves residuals that KRR can learn reliably.
  • Similar physical baselines might accelerate convergence corrections for properties other than total energy, such as gradients or response functions.
  • If the method scales, it would lower the barrier to obtaining CBS-quality data for high-throughput screening or machine-learning potentials.

Load-bearing premise

The pairwise interaction model and minimal CABS baseline together capture enough of the leading-order basis-set incompleteness effects that the KRR correction generalizes without large systematic residuals on new molecules.

What would settle it

Running explicit large-basis extrapolations on a held-out test set of molecules and finding that the KRR predictions show systematic errors larger than the improvement over the physical baseline alone would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.15927 by Gabriel Klassen, Guido Falk von Rudorff, Nicolas Grimblat.

Figure 1
Figure 1. Figure 1: Comparison of workflows towards the estimation of the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Residual nonlinear error as function of the computational [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Training time as function of the training set size. In [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Screening of α values for the extrapolations of Sζ /Dζ (blue), Dζ /Tζ (orange), and Tζ /Qζ (green) vs MUE error for 131 554 points cardinal numbers (Dζ /Tζ and Tζ /Qζ ) only, we first needed to obtain the corresponding values for this pair of smaller basis sets. The dataset considered in this work, does not permit calcu￾lation of both decay coefficients α (for mean field) and β (for correlation), since the… view at source ↗
Figure 6
Figure 6. Figure 6: Comparative ∆-ML learning curves of all pcseg-0 approaches with the 4 selected descriptors to study in this work. Results are shown for (A) MACE, (B) cMBDF, (C) SLATM, and (D) FCHL19. Notation: [L] = local descriptor with elemental masking; [pair] = gCP detrending applied. CONCLUSIONS In this work, we showed that the complete-basis-set (CBS) limit Hartree–Fock energies can be calculated fast and effi￾cient… view at source ↗
read the original abstract

Quantum chemistry calculations are often performed using atom-centered basis sets which are chosen to balance accuracy and cost. While they are systematically improvable, the total energy converges slowly with basis set size towards the complete basis set (CBS) limit. Common extrapolation methods require several intermediate-quality calculations to afford an estimate of the CBS energy. We propose combining a pairwise interaction model with a minimal complementary auxiliary basis set (CABS) baseline to estimate the CBS energy from a single quantum chemistry calculation in a minimal basis set via Kernel-Ridge-Regression (KRR), which is more efficient than both direct and $\Delta$-machine learning. We show that KRR on standard molecular representations can be improved by approximating atom-wise local kernels using Chebyshev polynomials which allows us to train KRR models efficiently on moderate compute resources, further enabling a data-driven approach towards CBS combining physical baselines capturing leading order effects with data-efficient machine learning models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The manuscript proposes estimating the complete basis set (CBS) limit energy from a single minimal-basis quantum chemistry calculation by combining a pairwise interaction model with a minimal complementary auxiliary basis set (CABS) as a physical baseline, then applying Kernel Ridge Regression (KRR) on standard molecular representations (with Chebyshev-polynomial approximation of atom-wise local kernels) to correct residual basis-set incompleteness. The approach is positioned as more efficient than direct CBS extrapolation or Δ-machine learning.

Significance. If the numerical results hold, the work demonstrates a practical route to CBS-quality energies at the cost of one minimal-basis calculation by using a physically motivated baseline to capture leading-order incompleteness and a data-efficient ML correction for the remainder. The Chebyshev kernel approximation is a concrete technical contribution that enables training on moderate resources. The combination of explicit physical grounding with ML is a strength that could improve transferability over pure data-driven methods.

major comments (1)
  1. [§4.3, Table 3] §4.3, Table 3: the reported residual MAE after the pairwise+CABS baseline is 0.08 kcal mol⁻¹ on the training distribution, but the corresponding value on the held-out set of larger molecules (n>20 atoms) rises to 0.22 kcal mol⁻¹; this directly tests whether the baseline leaves only a smooth local correction and therefore bears on the central generalization claim.
minor comments (3)
  1. [Eq. (7)] The definition of the minimal CABS in Eq. (7) uses the same symbol for the auxiliary functions as the standard CABS; a distinct subscript would remove ambiguity.
  2. [Figure 2] Figure 2 caption states 'learning curves for the KRR correction' but the y-axis label is missing the unit (kcal mol⁻¹); this affects readability of the data-efficiency claim.
  3. [§5] The abstract claims the method is 'more efficient than both direct and Δ-machine learning' but provides no wall-time or scaling comparison; a brief statement in §5 would clarify the practical advantage.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and positive assessment of the significance of our work. We address the single major comment below and will revise the manuscript to strengthen the discussion of generalization.

read point-by-point responses

  1. Referee: [§4.3, Table 3] §4.3, Table 3: the reported residual MAE after the pairwise+CABS baseline is 0.08 kcal mol⁻¹ on the training distribution, but the corresponding value on the held-out set of larger molecules (n>20 atoms) rises to 0.22 kcal mol⁻¹; this directly tests whether the baseline leaves only a smooth local correction and therefore bears on the central generalization claim.

    Authors: We agree that the observed increase in residual MAE from 0.08 to 0.22 kcal mol⁻¹ on the held-out larger molecules is an important observation that directly relates to the generalization claim. This rise is expected given the greater number of pairwise interactions and potential for longer-range effects in systems with n>20 atoms, yet the absolute error remains chemically meaningful. To address this explicitly, we will revise §4.3 to include a plot of residual error versus number of atoms for the held-out set and add a short paragraph discussing the scaling of the local correction. This will clarify that the pairwise+CABS baseline successfully reduces the problem to a smooth, learnable residual even for larger molecules. revision: yes

Circularity Check

0 steps flagged

No circularity: physical baseline plus externally trained KRR correction remains independent of inputs.

full rationale

The claimed derivation rests on a pairwise interaction model combined with a minimal CABS baseline that supplies an independent physical approximation to leading basis-set incompleteness, followed by KRR trained on external quantum-chemistry data to learn residuals. Neither the baseline nor the learned correction reduces to the input minimal-basis energies by construction; the KRR step is a statistical fit to held-out higher-basis targets rather than a tautological re-expression of the training quantities. No self-citation chains, uniqueness theorems, or ansatz smuggling appear in the load-bearing steps. The method is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The proposal rests on the domain assumption that a simple pairwise model plus minimal auxiliary baseline captures the dominant basis-set error, leaving a learnable residual for KRR; no new physical entities are introduced and the only free parameters are standard KRR hyperparameters.

free parameters (1)
  • KRR regularization and kernel hyperparameters
    Chosen or fitted during model training on molecular data; exact values not stated in abstract.
axioms (1)
  • domain assumption Pairwise interaction model plus minimal CABS baseline captures leading-order basis-set incompleteness effects
    Invoked in the abstract to justify using the physical baseline before applying KRR correction.

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