An Accurate and Transferable Machine Learning Potential for Carbon

Patrick Rowe; Volker L Deringer; Piero Gasparotto; Gábor Csányi; Angelos Michaelides

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An Accurate and Transferable Machine Learning Potential for Carbon

Patrick Rowe, Volker L Deringer, Piero Gasparotto, Gábor Csányi, Angelos Michaelides

arXiv:2006.13655v1Jun 24, 2020physics.comp-phcond-mat.mtrl-sci

TL;DR

Carbon’s extreme structural diversity requires an interatomic potential that is both accurate near crystalline minima and transferable across amorphous, liquid, defect, and nanostructured environments. The paper develops GAP-20 by combining broad structural training data, dispersion-aware DFT references, long-range pair interactions, and optimised SOAP descriptors. GAP-20 achieves meV-scale crystalline accuracy while accurately treating defects, surfaces, and liquid carbon across diverse conditions.

Problem
Carbon’s many allotropes and interaction regimes make it difficult for one potential to combine high numerical accuracy with broad transferability.
Method
GAP-20 combines expanded carbon training databases, dispersion-corrected DFT references, 2b/3b/many-body GAP components, and optimised SOAP descriptors.
Results
GAP-20 predicts key crystalline formation energies to a few meV, accurately describes surfaces and defects, and handles high-temperature liquid carbon across temperatures and densities.
Takeaways & Limitations
The model is suitable as a general-purpose carbon ML potential for diverse bulk and nanostructured applications.
Takeaways & Limitations
Hydrogenated and oxidised carbon are outside the scope, while long-range van der Waals interactions are treated only approximately.

Abstract

from arXiv · show

We present an accurate machine learning (ML) model for atomistic simulations of carbon, constructed using the Gaussian approximation potential (GAP) methodology. The potential, named GAP-20, describes the properties of the bulk crystalline and amorphous phases, crystal surfaces and defect structures with an accuracy approaching that of direct ab initio simulation, but at a significantly reduced cost. We combine structural databases for amorphous carbon and graphene, which we extend substantially by adding suitable configurations, for example, for defects in graphene and other nanostructures. The final potential is fitted to reference data computed using the optB88-vdW density functional theory (DFT) functional. Dispersion interactions, which are crucial to describe multilayer carbonaceous materials, are therefore implicitly included. We additionally account for long-range dispersion interactions using a semianalytical two-body term and show that an improved model can be obtained through an optimisation of the many-body smooth overlap of atomic positions (SOAP) descriptor. We rigorously test the potential on lattice parameters, bond lengths, formation energies and phonon dispersions of numerous carbon allotropes. We compare the formation energies of an extensive set of defect structures, surfaces and surface reconstructions to DFT reference calculations. The present work demonstrates the ability to combine, in the same ML model, the previously attained flexibility required for amorphous carbon [Phys. Rev. B, 95, 094203, (2017)] with the high numerical accuracy necessary for crystalline graphene [Phys. Rev. B, 97, 054303, (2018)], thereby providing an interatomic potential that will be applicable to a wide range of applications concerning diverse forms of bulk and nanostructured carbon.

1 Introduction

Carbon’s structural and property diversity makes accurate, transferable atomistic modelling difficult. GAP-20 combines broad flexibility with high accuracy across crystalline, amorphous, defect, surface, and liquid-carbon settings.

Carbon spans zero- to three-dimensional allotropes with metallic, semiconducting, and insulating phases and exceptional mechanical properties.
Existing empirical potentials have inherent limitations, particularly when modelling structures departing from ideal diamond and graphite.
GAP-17 provided transferability across diverse amorphous and liquid environments but accepted a degree of numerical error.
GAP-20 targets subtle nanostructure and defect formation energies and phonon dispersions to within meV accuracy while retaining GAP-17’s flexibility and transferability.Its reference data use dispersion-corrected DFT to account for longer-range interactions in low-dimensional carbon structures.
GAP-20 predicts formation energies of diamond, graphite, fullerenes, and nanotubes to a few meV, with comparable accuracy for crystalline and amorphous surfaces.Defect formation-energy errors are significantly lower than those of comparable empirical models, and liquid carbon is accurately predicted across temperatures and densities.

2 Generation and Selection of Training Data

The training set is designed to cover carbon’s diverse, physically relevant descriptor space without exhaustively sampling the full chemical configuration space. Farthest point sampling is supplemented with chemically selected configurations to improve local accuracy.

Carbon’s training problem spans crystalline phases, amorphous forms, graphene, nanotubes, and fullerenes across differing dimensionalities.
Physically relevant carbon configurations occupy a reduced subset of the full 3N-dimensional space, motivating sampling in descriptor space.SOAP descriptors represent local atomic environments rather than global structures.
Reference energies, forces, and virial stresses are computed consistently with tightly converged plane-wave DFT including dispersion corrections.
The database combines earlier GAP-17 and graphene datasets with many new configurations covering crystalline phases and nearby phase-space regions.The resulting database contains approximately 17,000 configurations with 1–240 atoms per cell.
Farthest point sampling selects structurally diverse configurations by maximising kernel distance, then mandatory chemically chosen structures restore dense sampling near important minima.The final database unites 4,000 FPS-selected points with the existing GAP-17 dataset.
The sketch-map uses kernel similarity to place similar configurations closer together and dissimilar configurations farther apart.It visualises varied sp1, sp2, and sp3 coordination environments across the training set.

3 Training of the Potential

GAP-20 decomposes the carbon potential into short- and long-range pair interactions plus higher-order many-body terms. Descriptor and sparsification tests select settings that balance accuracy, physical coverage, and evaluation cost.

GAP-20 represents the potential-energy surface using two-body, three-body, and high-dimensional many-body contributions.
The two-body term models short-range exchange repulsion and bonding, then transitions beyond 4.0 Å to an analytical r^-6 dispersion tail up to 10 Å.
The three-body and SOAP components represent higher-order bonding contributions, with SOAP encoding local environments through invariant neighbour-density representations.
A 4.5 Å SOAP cutoff is selected despite a force-error minimum at 2.9 Å because graphitic structures require information from layers separated by approximately 3.3 Å.
Force errors decrease rapidly to approximately 1,500 sparse points and then level off, while the chosen 9,000 points are very tightly converged.
SOAP optimisation finds that radial basis expansion affects convergence more than angular expansion, motivating a radial-biased choice.The figure evaluates force error and relative computational cost across cutoff, sparse-point count, and basis-set order.

4 Crystalline Carbon

GAP-20 accurately models crystalline carbon structures and energetics while incorporating long-range interactions needed for layered and nanostructured phases. Its phonon predictions remain accurate across diamond, graphene, and nanotubes despite broader transferability than earlier specialized models.

Static properties: 0.2% average lattice-parameter error compares with 5% for Tersoff, 0.3% for LCBOP, 4% for REBO-II, and 1% for AIREBO.Short-cutoff models particularly mispredict graphite inter-layer spacings, whereas long-range van der Waals terms improve this behavior.
Energetics: GAP-20 predicts tested allotrope atomisation energies within 1%, including subtle cubic–hexagonal diamond differences and nanotube and fullerene energetics.Agreement with optB88-vdW DFT is excellent when both the isolated atom and graphite are used as reference states.
Energetics: Graphite formation-energy comparisons improve substantially when energies are referenced to each model’s graphite state, removing offsets dominated by isolated-atom treatment.The figure contrasts atomisation energies against the isolated gas-phase atom with formation energies relative to graphite.
Lattice dynamics: Accurate phonon dispersions provide access to lattice dynamics and thermodynamically relevant properties such as thermal expansion and constant-volume heat capacity.The phonon spectrum is experimentally measurable and directly probes finite-temperature lattice behavior.
Lattice dynamics: GAP-20 phonon dispersions are accurate for diamond, graphene, and nanotubes; graphene frequencies are correct within 4 meV versus 1 meV for the earlier graphene-only model.The broader model preserves comparable graphene dispersion quality while also describing diamond and two nanotube chiralities.

5 Surfaces of Carbon

Surfaces are challenging because their description involves competing physical interactions. GAP-20 predicts surface energies and relaxed surface structures accurately across several crystalline and amorphous carbon surfaces.

Surface modelling is challenging because surfaces involve competing physical interactions.
For amorphous surfaces, surface energies are reported only in the as-cut configuration because surface relaxation is extensive.The surface energy calculation uses slabs containing two surfaces and compares their energy with bulk reference energies.
GAP-20 predicts diamond surface energies within 7%, except for relaxed diamond (111), where the error is 15%.Relaxed surface atom positions agree with reference structures to an average error of 10^-3 Å.
GAP-20 predicts graphite (0001) surface energies within 3 meV Å^-2, corresponding to a 20% error.The very small graphite surface energy makes this surface particularly difficult to model.
GAP-20 generally outperforms empirical potentials across diamond surfaces, which do not show uniformly low errors.For graphite-layer binding, Tersoff and REBO-II predict no binding, causing spontaneous exfoliation in simulations.

6 Defective Carbon

Carbon defects strongly affect material properties, while their diverse reconstructions make accurate modelling difficult. GAP-20 generally reproduces defect energetics and geometries, with larger errors for some diamond and nanotube cases.

Defects can strongly affect carbon materials’ structural, optical and thermal properties, and carbon rehybridisation enables diverse stabilising reconstructions.
GAP-20 predicts most defect formation energies within 10%, with especially accurate Stone-Wales defects in graphite and graphene.Diamond defect errors range from 25–35%, while defective nanotube errors range from 0–11%.
Defect formation energies are sensitive to the SOAP cutoff, selected training data and number of sparse points.
GAP-20 reproduces most defect geometries with atomic position errors below 10^-2 Å relative to DFT structures.It captures graphene Stone-Wales buckling, nanotube distortions, graphene divacancy reconstructions and diamond defect geometries.
GAP-20 predicts graphene monovacancies symmetrically rather than reproducing their Jahn–Teller asymmetry.The symmetric and asymmetric geometries differ by approximately 350 meV, and typical atomic-position errors remain about 0.1 Å in these inaccurate cases.

7 Liquid Carbon

Liquid carbon samples diverse local environments across broad temperature and density ranges, providing a demanding transferability test. GAP-20 agrees well with DFT for liquid structure while retaining accuracy for crystalline and defect properties.

Liquid simulations test potential flexibility because they explore many local configurations across broad densities and temperatures.The study examines densities from 1.5–3.5 g cm^-3 at 5000 K and temperatures from 5000–9500 K at fixed density.
GAP-20 shows very good agreement with ab initio data for liquid-carbon radial and angular distribution functions across the studied temperatures.
GAP-20 captures increased liquid structuring as temperature decreases from 9500 to 4500 K.Below approximately 3500 K, it predicts amorphous-glass formation followed by slow graphitisation.
GAP-20 captures changes in liquid-carbon bonding with density, including increased sp1 character at very low densities.The sp1 increase is reflected in bond angles close to 180 degrees.
The potential models liquid structures over wide conditions while also predicting phonon relations and defect formation energies accurately.This spans energy scales from meV differences between structures to much larger fluctuations encountered in liquid simulations.

8 Transferability of the Potential

GAP-20 is tested beyond its training set using random structure searches and transformations, with comparisons against DFT assessing transferability in unfamiliar, high-energy regions. It reproduces energies and transformation barriers well while avoiding several qualitative errors seen in empirical potentials.

Transferability tests: GAP-RSS tests transferability by exploring high-energy regions of the potential energy surface with structures not explicitly included in training.The search is paired with DFT energy comparisons and transformation pathways absent from the training data.
Random structure search: Across all 1000 random-search structures, GAP-20 energies agree well with the reference DFT energies.The authors contrast this with empirical models that can generate qualitatively incorrect low-energy structures.
Random structure search: GAP-RSS identifies AB-stacked graphite as the lowest-energy allotrope and correctly ranks AA- and ABC-stacked graphite higher.It also finds diamond, lonsdaleite, and several exotic allotropes absent from the training dataset.
Random structure search: The eight-atom periodic search mostly samples bulk amorphous and crystalline structures, with few fullerene or nanotube configurations.A structurally distinct cluster of highly sp1-rich structures appears outside the main training-data regions.
Transformation tests: For graphene bond rotation and C60 rotation, GAP-20 reproduces the DFT barrier height and shape more accurately than the tested empirical potentials.LCBOP and Tersoff overestimate the graphene rotation by more than 30 eV, while several empirical models introduce spurious minima.

9 Conclusion

The paper presents GAP-20 as a broadly applicable carbon potential that accurately treats diverse phases, defects, surfaces, liquids, and transformations relative to DFT. Its broad scope entails higher cost than empirical potentials, limited treatment of chemically modified carbon, and a trade-off between breadth and property-specific accuracy.

Conclusion: GAP-20 accurately treats diverse carbon structures and properties, including bulk phases, defects, surfaces, liquid carbon, and bond-breaking or bond-forming processes.The authors validate performance against reference DFT across different phases wherever possible.
Conclusion: Diagnostic random searches and transformations not included in training suggest that GAP-20 is transferable beyond its explicitly sampled configurations.These tests support the model’s application to structures and processes outside the training set.
Limitations: GAP-20 is significantly more affordable than direct ab initio simulation but substantially more expensive to evaluate than empirical potentials.Empirical potentials therefore retain access to even larger-scale systems.
Limitations: Hydrogenated and oxidised carbon are outside the current scope, while long-range van der Waals interactions are treated only approximately.Adding other elements would require additional data and interactions involving charge rearrangements and long-range electrostatics.
Conclusion: The total training dataset is freely available, and users can retrain the potential by changing the selected training configurations for a targeted application.The authors frame this as a way to pursue higher accuracy in a particular region of interest.

10 Supplementary Material

Supplementary material provides detailed results and additional methodological information for evaluating and using GAP-20. It also documents hyperparameters, force errors, computational cost, and dispersion-related tests omitted from the main text.

Supplementary contents: The supplementary material contains full formation energies and phonon dispersion curves for all tested models.
Supplementary contents: Additional material covers GAP hyperparameter selection and command-line arguments.
Supplementary contents: Further supplementary analyses report graphene bilayer separation curves, force errors, optimisation details, and the computational cost relative to DFT.