∆-Machine Learning to Elevate DFT-based Potentials and a Force Field to the CCSD(T) Level - Illustrated for Ethanol

Aug 5, 2026

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Progress in machine learning enables potentials that combine first-principles accuracy with dramatically faster evaluation. Recently, “∆-machine learning” has been used to upgrade low-level density functional theory (DFT) potential energy surfaces (PESs) toward CCSD(T) benchmark accuracy. The work validates this strategy across molecule sizes and then tests its generality for PBE, M06, M06-2X, and PBE0+MBD using ethanol. Linear regression with permutationally invariant polynomials fits low-level and correction PESs, followed by RMSE and fidelity checks for stationary points, vibrational modes, and torsional energetics. Results show comparable improvements across functionals and additional gains for DFT gradients without coupled-cluster gradient correction, plus initial force-field correction observations.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages42

Reading Time1 h 46 min

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Abstract

Introduction

Machine learning potentials and PES fitting

∆-machine learning versus other correction strategies

Bottlenecks in high-level CCSD(T) calculations

arXiv :2407 .20050v1 [physics .chem-ph] 29 Jul 2024 ∆-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∗ ,‡ †Department of Physics and Materials Science, University of Luxembourg, L-1511, Luxembourg City, Luxembourg. ‡Department of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory University, Atlanta, Georgia 30322, U. S.A. ¶Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, U. S.A. and Department of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332, U.S.A §Independent Researcher, Toronto, Ontario M9B0E3, Canada ∥Department of Chemistry, Fudan University, Shanghai, 200438, P. R. China ⊥Dipartimento di Chimica, Universit`a degli Studi di Milano, via Golgi 19, 20133 Milano, Italy E-mail: [apurba.nandi@uni.lu](apurba.nandi@uni.lu) ; [alexandre.tkatchenko@uni.lu](alexandre.tkatchenko@uni.lu) ; [jmbowma@emory.edu](jmbowma@emory.edu) 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,“∆-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 H3 O+ 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 datasets, 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 five atoms, based on fitting thousands of CCSD(T) energies 1–5 or forces. 6,7 Some of these ML approaches have been using 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 dataset for that method was generated using 500 K direct dynamics based on the PBE0 functional. CCSD(T) datasets for larger molecules are rare, and the 10-atom formic acid dimer is one prominent recent example; PESs for this dimer have been reported using PIPs 15 and later an atom-centered high-dimensional NN. 16 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. 17,18 The PIP-based automated ROBOSURFER software

FAQs

What is ∆-machine learning in the context of improving potential energy surfaces?

∆-machine learning adds a correction learned from lower-level quantities (such as DFT energies and gradients) to elevate a PES toward CCSD(T) benchmark quality.

How is generality tested beyond a single functional?

The approach is evaluated using multiple DFT functionals (PBE, M06, M06-2X, and PBE0+MBD) with ethanol as the example molecule, using fitted low-level and correction PESs.

What metrics are used to assess improvement and fidelity?

Training and test performance are compared using RMSE, and fidelity is further examined via energetics of stationary points, normal mode frequencies, and torsional potentials.

Aug 5, 2026

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