Δ-Machine Learning to Elevate DFT-Based Potentials and a Force Field to the CCSD(T) Level Illustrated - Article

Aug 5, 2026

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Progress in machine learning enables potential energy surfaces that combine first-principles accuracy with substantially faster evaluation. The Δ-machine learning strategy elevates low-level density functional theory (DFT) energies and gradients to coupled cluster CCSD(T) quality by learning correction surfaces. The work demonstrates transferability across molecules from H3O+ to 15-atom systems for several functionals, using ethanol as an example. Permutationally invariant polynomial regression supports fidelity tests on energetics, vibrational spectra, and torsional potentials, with noted gains even when coupled-cluster gradients are unavailable, and includes results for correcting a molecular mechanics force field.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages13

Reading Time33 min

AccessFree to download

Introduction

Machine-learning PES development and Δ-ML motivation

Low-level to high-level correction strategy (DFT to CCSD(T))

Model accuracy evaluation framework

This article is licensed under CC-BY 4.0 [pubs.acs.org/JCTC](pubs.acs.org/JCTC) Article Δ‑Machine Learning to Elevate DFT-Based Potentials and a Force Field to the CCSD(T) Level Illustrated for Ethanol Apurba Nandi, * Priyanka Pandey, Paul L. Houston, Chen Qu, Qi Yu, Riccardo Conte, Alexandre Tkatchenko, * and Joel M. Bowman* Cite This: J. Chem. Theory Comput. 2024, 20, 8807−8819 Read Online Downloaded via UNIV OF MILAN on November 7, 2024 at 19:49:51 (UTC) . See [https://pubs.acs.org/sharingguidelines](https://pubs.acs.org/sharingguidelines) for options on how to legitimately share published articles. ACCESS Metrics & More Article Recommendations ABSTRACT: Progress in machine learning has facilitated the development of potentials that offer both the accuracy of first-principles techniques and vast increases in the speed of evaluation. Recently, Δ-machine learning has been used to elevate the quality of a potential energy surface (PES) based on low-level, e.g., density functional theory (DFT) energies and gradients to close to the gold-standard coupled cluster level of accuracy. We have demonstrated the success of this approach for molecules, ranging in size from H3O+ to 15-atom acetyl-acetone and tropolone. These were all done using the B3LYP functional. Here, we investigate the generality of this approach for the PBE, M06, M06-2X, and PBE0 + MBD functionals, using ethanol as the example molecule. Linear regression with permutationally invariant polynomials is used to fit both low-level and correction PESs. These PESs are employed for standard RMSE analysis for training and test data sets, and then general fidelity tests such as energetics of stationary points, normal-mode frequencies, and torsional potentials are examined. We achieve similar improvements in all cases. Interestingly, we obtained significant improvement over DFT gradients where coupled cluster gradients were not used to correct the low-level PES. Finally, we present some results for correcting a recent molecular mechanics force field for ethanol and comment on the possible generality of this approach. ■ INTRODUCTION Developing high-dimensional, ab initio-based potential energy surfaces (PESs) is an active area of theoretical and computational research. Major progress has been made in using and developing machine learning (ML) approaches for PESs with more than four atoms, based on fitting thousands of CCSD(T) energies1−5 or forces.6,7 Some of these ML approaches have used permutationally invariant polynomials (PIPs) or PIPs as inputs to neural network software. 1−5 Of course, there are numerous other ML methods. It is perhaps of interest and relevance to this paper that the precision of a PIP PES for ethanol was shown to be as good as the best performing ML methods and to be substantially faster (factors of 10 or more)8 than all the ML methods considered, i.e., GAP-SOAP,9 ANI,10 DPMD, 11 sGDML,6,7 PhysNet, 12 KREG, 13 and pKREG.14 The data set for these comparisons was from the rMD17 database,15 which uses 500 K direct dynamics based on the PBE0 functional to obtain energies and forces. The metrics used in the “learning curves” were root-mean-square errors in energies and forces. This followed the standard protocol used earlier to assess many ML methods for potentials.16 CCSD(T) data sets for larger molecules are rare, ing to the steep scaling of CCSD(T) calculations of order N,7 N being the number of basis functions. The potential energy surface for the 10-atom formic acid dimer is one example where PESs have been reported at the CCSD(T) level, using PIPs 17 and later an atom-centered high-dimensional NN.18 Complex reactive potentials for 6 and 7-atom chemical reactions, which are fitted to tens of thousands or even hundred thousand CCSD(T) energies, have been reported.19,20 The PIP-based automated ROBOSURFER software5 has been applied to develop a number of complex PESs for 9-atom chemical reactions.21,22 Correcting ab i

FAQs

What does Δ-machine learning do in this study?

It learns corrections to low-level electronic-structure results (e.g., DFT energies and gradients) so the resulting potential energy surface approaches CCSD(T) accuracy.

Which molecule is used as the main example for testing generality?

Ethanol is used as the example to investigate whether the Δ-ML approach generalizes across multiple functionals.

How is the quality of the learned potential energy surfaces evaluated?

The study performs RMSE analyses on training and test sets and conducts fidelity checks using stationary-point energetics, normal-mode frequencies, and torsional potentials.

Aug 5, 2026

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