PaperlayerView exact revision on arXiv

Source-linked AI summary

A Fourth-Generation High-Dimensional Neural Network Potential with Accurate Electrostatics Including Non-local Charge Transfer

Tsz Wai Ko, Jonas A. Finkler, Stefan Goedecker, Jörg Behler

arXiv:2009.06484v1cond-mat.mtrl-sciphysics.chem-phphysics.comp-ph

TL;DR

Existing machine-learning potentials struggle with long-range charge transfer and multiple charge states because they rely largely on local properties. This paper introduces a fourth-generation potential combining environment-dependent electronegativities with accurate atomic energies, achieving improved agreement with electronic-structure calculations across representative systems.

  • Problem

    Existing machine-learning potentials inadequately describe long-range charge transfer, long-range electrostatics, and systems with different charge states.

  • Method

    The fourth-generation potential combines accurate short-range atomic energies with untruncated electrostatics and charge equilibration based on environment-dependent neural-network electronegativities.

  • Results

    4G-HDNNPs correctly describe representative systems, improving agreement of energies, forces, and charges with DFT while identifying structures missed by earlier generations.

  • Takeaways & Limitations

    The method extends machine-learning potentials to chemical and materials-science scenarios involving long-range charge transfer and multiple charge states.

Abstract

from arXiv · show

Machine learning potentials have become an important tool for atomistic simulations in many fields, from chemistry via molecular biology to materials science. Most of the established methods, however, rely on local properties and are thus unable to take global changes in the electronic structure into account, which result from long-range charge transfer or different charge states. In this work we overcome this limitation by introducing a fourth-generation high-dimensional neural network potential that combines a charge equilibration scheme employing environment-dependent atomic electronegativities with accurate atomic energies. The method, which is able to correctly describe global charge distributions in arbitrary systems, yields much improved energies and substantially extends the applicability of modern machine learning potentials. This is demonstrated for a series of systems representing typical scenarios in chemistry and materials science that are incorrectly described by current methods, while the fourth-generation neural network potential is in excellent agreement with electronic structure calculations.

I. INTRODUCTION

Machine-learning potentials offer quantum-mechanical accuracy with empirical-potential efficiency, but local approaches miss long-range interactions, non-local charge transfer, and different charge states. The work introduces a fourth-generation high-dimensional neural network potential combining environment-dependent electronegativities and accurate short-range atomic energies to address these limitations.

  • Motivation: ML potentials combine the accuracy of quantum-mechanical electronic-structure calculations with the efficiency of simple empirical potentials.They have emerged as an alternative approach for constructing potential-energy surfaces.
  • Limitations: 2G and 3G high-dimensional neural network potentials rely on local descriptions and therefore cannot distinguish systems whose global electronic structures differ.In the illustrated molecular system, identical local environments can correspond to different distant-atom-dependent charges.
  • Limitations: 2G potentials neglect long-range interactions, including electrostatics beyond the cutoff radius and often dispersion interactions, creating potential errors particularly for ionic systems.These interactions may substantially accumulate in condensed systems.
  • Limitations: 3G potentials remain unable to describe long-range charge transfer and different charge states because atomic partial charges depend only on local chemical environments.Neglecting non-local charge transfer and global charge-distribution changes can cause qualitative failures.
  • Contribution: The proposed 4G HDNNP combines accurate short-range atomic energies with charge equilibration based on environment-dependent electronegativities.Both atomic energies and electronegativities are represented by atomic neural networks as functions of chemical environments, and the method is illustrated on challenging chemistry and materials-science model systems.

II. METHOD

The 4G-HDNNP combines short-range atomic energies with untruncated long-range electrostatics, obtaining charges through environment-dependent electronegativities and global charge equilibration. Its charges and short-range energies incorporate non-local electronic information, enabling non-local charge transfer while avoiding electrostatic double counting.

  • Energy decomposition: The total energy combines a short-range atomic-energy contribution with an untruncated long-range electrostatic contribution.The electrostatic energy is computed from atomic charges, while the short-range energy is a sum of atomic energies.
  • Charge equilibration: Atomic neural networks predict environment-dependent electronegativities, which enter charge equilibration to determine atomic charges.Neighbor positions within cutoff radius Rc are encoded by atom-centered symmetry functions, preserving translational, rotational, and permutational invariance.
  • Charge equilibration: Global charge equilibration makes charges depend on atomic positions, chemical composition, and total system charge, naturally including non-local charge transfer.The charges are trained directly against DFT reference charges, unlike in CENT, where they are intermediate quantities optimized only through DFT energy errors.
  • Short-range energy: Short-range atomic energies use local chemical environments together with equilibrated partial charges, capturing changes in local electronic structure without double counting electrostatic energy.The short-range networks are trained on energies and forces to represent the remaining part of the reference energy.

III. RESULTS AND DISCUSSION · Overview

The results demonstrate the limitations of local-property machine-learning potentials across diverse non-periodic and periodic chemical and materials systems. The 4G-HDNNP overcomes these limitations, reproducing distant-dopant-induced adsorption-geometry changes with potential-energy surfaces close to DFT.

  • Overview: The study tests local-property ML potentials and 4G-HDNNP on non-periodic and periodic systems spanning typical chemistry and materials-science scenarios.The non-periodic set includes a covalent organic molecule, a small metal cluster, and an ionic-material cluster.
  • Overview: The selected non-periodic systems represent substantially different types of atomic interactions.They comprise a covalent organic molecule, a small metal cluster, and a cluster of an ionic material.
  • Overview: The demonstrated systems cover typical situations relevant to chemistry and materials science.The paper frames these examples as representative of a wide range of non-periodic and periodic systems.
  • Overview: The results address applications in heterogeneous catalysis.The catalytic example examines adsorption geometry changes caused by dopant atoms introduced far from a cluster.
  • Overview: The 4G-HDNNP reproduces the change in adsorption geometry when dopant atoms are introduced far away from the cluster in the slab.This behavior contrasts with established ML potentials based on local properties.
  • Overview: In all examined cases, the 4G-HDNNP potential-energy surface is very close to the DFT results.Figure 3 compares DFT partial charges with unscaled and scaled 3G-HDNNP charges and 4G-HDNNP charges for C10H2 and C10H+ 3.

Non-Periodic Cases · A Benchmark for Organic Molecules

The C10 carbon-chain benchmark shows that local HDNNPs cannot reliably distinguish neutral and charged molecules or reproduce their non-local charge redistribution. The 4G-HDNNP captures the resulting long-range electrostatics, charges, energies, and forces substantially more accurately than earlier generations.

  • A Benchmark for Organic Molecules: A ten-carbon sp-hybridized chain terminated by hydrogen atoms provides a benchmark for comparing neutral C10H2 and protonated C10H3+ molecules.The protonated molecule changes terminal-carbon hybridization from sp to sp2.
  • A Benchmark for Organic Molecules: Local structural information alone cannot make existing potentials simultaneously applicable to neutral and charged systems.The limitation arises from lacking explicit global charge distribution and total-charge information.
  • A Benchmark for Organic Molecules: Protonation modifies the electronic structure and DFT charges throughout the molecule, including at distances far from the added proton.This non-local response distinguishes the neutral and cationic molecules despite their similar local environments in the remote region.
  • A Benchmark for Organic Molecules: The 4G-HDNNP matches DFT atomic charges very accurately for both molecules because it can distinguish their global charge states.It naturally includes the correct long-range electrostatic energy for global charges represented in the training set.
  • A Benchmark for Organic Molecules: The 3G-HDNNP assigns qualitatively incorrect averaged charges where nearly identical environments correspond to different atomic charges.Contradictory training information for remote atoms produces high fitting errors and averaged predictions unlike the DFT references.
  • A Benchmark for Organic Molecules: Across total energies, atomic charges, and forces, the 4G-HDNNP has much lower errors than the 2G- and 3G-HDNNPs.The inaccurate charges of the 3G-HDNNP and the short-range-only 2G-HDNNP degrade the potential energy surface.
  • A Benchmark for Organic Molecules: 0.037 eV/Å is the 4G-HDNNP average force error for all atoms in both molecules, whereas 2G-HDNNP force errors are substantially larger in remote regions.The 2G model performs better only near the extra proton, where it appears as a local structural feature.
  • A Benchmark for Organic Molecules: <1 meV/atom is the 2G-HDNNP energy error for both molecules despite its relatively high force errors.The result suggests error compensation between atomic energies, particularly between the two molecular halves in C10H3+.

Metal Clusters: Ag3

Ag3 provides a stringent test because different total charges produce different ground states, eliminating the unique position–energy relation required by a local 2G-HDNNP. By incorporating total charge and resulting partial charges, the 4G-HDNNP accurately predicts both charged-cluster minima and reduces energy errors by several orders of magnitude.

  • Charge-state dependence: Ag3 is studied in two charge states whose ionization-dependent potential-energy surfaces can have different ground states.Because the cluster is small, each atomic environment contains the full system, so long-range effects are absent.
  • Charge-state dependence: Combining Ag3+ and Ag3− data removes the unique relation between atomic positions and energy, preventing a 2G-HDNNP from distinguishing the clusters.The training-set ambiguity arises because the total charge differs while the structural information alone is insufficient.
  • 4G-HDNNP performance: The 4G-HDNNP uses structural information together with total charge and resulting partial charges to predict both minima and atomic partial charges with very high accuracy.The DFT-optimized minima and atomic partial charges are used as the reference for comparison.
  • 4G-HDNNP performance: 1.166 meV/atom and 0.320 meV/atom are the 4G-HDNNP energy errors for Ag3− and Ag3+, respectively, versus 0.605 and 2.017 eV/atom for the 2G-HDNNP.The 2G-HDNNP errors are several orders of magnitude larger than the corresponding 4G-HDNNP errors.

NaCl Cluster Ions

The Na9Cl+ 8 cluster demonstrates that distant structural changes alter the electronic structure and potential-energy surface, which 4G-HDNNP captures accurately while 2G- and 3G-HDNNPs miss important qualitative differences. Its forces likewise differ between cluster ions and are well reproduced by 4G-HDNNP.

  • System construction: Removing a neutral sodium atom from the positively charged Na9Cl+ 8 cluster creates a second cluster ion with fixed remaining-atom positions.The initial structure was obtained by DFT geometry optimization, and the potential-energy surfaces were compared while moving one blue sodium atom along a one-dimensional path.
  • Energy curves: More than 0.1 ˚A separates the DFT minima, showing that a distant brown sodium atom changes the blue atom’s equilibrium position.The 2G-HDNNP cannot distinguish these minima, whereas 4G-HDNNP identifies them at the right positions.
  • Energy curves: About 1 meV per atom is the deviation of the 2G- and 3G-HDNNP curves from DFT despite their qualitatively wrong minima, while 4G-HDNNP matches the corresponding DFT curves accurately.The small deviation is described as typical of state-of-the-art ML potentials but conceals the incorrect minima of the 2G- and 3G-HDNNPs.
  • Force curves: The DFT forces differ between both cluster ions and are well reproduced by 4G-HDNNP; 2G-HDNNP forces are exactly identical, whereas 3G-HDNNP forces differ slightly.The identical 2G-HDNNP forces result from the constant offset between energy curves, while 3G-HDNNP differences arise from additionally included long-range electrostatics.

Periodic Case · Au2 Cluster on MgO(001) · IV. SUMMARY

The Au2/MgO(001) case shows that distant Al doping changes charge transfer and adsorption stability, while the 4G-HDNNP captures these global effects. More broadly, the method combines charge equilibration with accurate atomic energies to describe long-range charge transfer, multiple charge states, and qualitatively correct potential-energy surfaces.

  • Periodic Case: Au2 clusters on MgO(001) are relevant to catalysis, including carbon monoxide oxidation, propylene epoxidation, water-gas-shift reactions, and hydrocarbon hydrogenation.The passage identifies supported gold clusters as systems studied for these catalytic reactions.
  • Au2 Cluster on MgO(001): Au2 on MgO(001) adopts either an upright non-wetting geometry attached to surface oxygen or a parallel wetting geometry spanning two Mg atoms.These are the two main adsorption geometries examined in the periodic case.
  • Au2 Cluster on MgO(001): Three Al dopants, corresponding to 2.86% of the slab, make the flat wetting geometry more stable than non-wetting, unlike pure MgO.The dopants were placed in the fifth layer, more than 10 Å from the gold atoms, yet changed the Au2 charge.
  • Au2 Cluster on MgO(001): The 2G-HDNNP predicts identical doped and undoped energy differences because the dopants lie outside the gold atoms’ local environments, whereas DFT and 4G-HDNNP agree on the doping-induced change.This demonstrates the limitation of a purely local description for the doped Au2/MgO(001) potential-energy surface.
  • Au2 Cluster on MgO(001): The 4G-HDNNP and DFT produce very similar non-wetting energy curves and resolve different equilibrium Au–O bond lengths for doped and undoped slabs.For the doped system, the reported bond lengths are 2.342 Å with 4G-HDNNP and 2.332 Å with DFT.
  • IV. SUMMARY: The 4G-HDNNP incorporates accurate long-range electrostatics, enabling treatment of long-range charge transfer and multiple charge states.This extends the method to chemical problems incorrectly described by local machine-learning potentials.
  • IV. SUMMARY: Environment-dependent atomic electronegativities from atomic neural networks drive charge equilibration, while the resulting global charges contribute to the potential energy.The approach combines CENT-like charge treatment with conventional second- and third-generation high-dimensional neural-network potentials.
  • IV. SUMMARY: Across systems where earlier HDNNPs give qualitatively wrong results, 4G-HDNNP improves agreement of energies, forces, and atomic charges with DFT and recovers previously missed local-minimum structures.The authors present the method as generally applicable to machine-learning potentials based on environment-dependent atomic energies and global charge distributions.

Simulation Details · Neural Network Potentials

The HDNNPs were built in RuNNer with consistent atom-centered representations and architectures across generations. Training, electrostatic screening, charge-density parameters, and geometry-optimization convergence were specified consistently across the studied systems.

  • Neural Network Potentials: HDNNPs were constructed using the freely available RuNNer program under the GPL3 license.
  • Neural Network Potentials: Atom-centered symmetry functions described atomic environments within 8-10 Bohr spatial cutoffs, depending on the system.
  • Neural Network Potentials: The same symmetry-function parameters and atomic neural-network architectures were used for different HDNNP generations within each system.
  • Neural Network Potentials: 90% of reference data trained the HDNNPs, while 10% formed an independent test set for reliability assessment and over-fitting detection.
  • Neural Network Potentials: Energies and forces trained the short-range atomic neural networks, with screened short-range Coulomb interactions facilitating the fitting of short-range energies and forces.
  • Neural Network Potentials: Electrostatic screening used inner and outer cutoffs of 1.69-2.54 ˚A and the symmetry-function cutoff, respectively.
  • Neural Network Potentials: Gaussian charge-density widths were set to the covalent radii of the elements.
  • Neural Network Potentials: Geometry optimizations used simple gradient descent with a force threshold of 10−4 Ha/Bohr ≈0.005 eV/˚A, matching the convergence used in validating DFT calculations.

DFT Calculations

DFT reference data were generated with all-electron FHI-aims using PBE, with convergence criteria, k-point sampling, spin treatments, charge analysis, and geometry optimization specified across the systems. Datasets were constructed from molecular dynamics, relaxation trajectories, and targeted structural sampling procedures.

  • Reference calculations: FHI-aims calculations used the PBE exchange-correlation functional with light settings, converging total energy, eigenvalue sums, and charge density to 10−5 eV, 10−2 eV, and 10−4 e, respectively.Au2-MgO used criteria tightened by a factor of 0.1.
  • Reference calculations: Au2-MgO calculations used a 3 × 3 × 1 k-point grid, while Au2-MgO, NaCl, and Ag3 systems received spin-polarized calculations.Reference atomic charges came from Hirshfeld population analysis, and geometries were optimized with the Broyden-Fletcher-Goldfarb-Shanno algorithm.
  • Dataset construction: C10H2/C10H+3 molecules and Ag3 clusters were sampled by Born-Oppenheimer molecular dynamics at 300 K for 5000 steps with a 0.5 fs timestep.Simulations used a Nosé-Hoover thermostat in the canonical ensemble with an effective mass of 1700 cm−1; relaxation trajectories were also added through force convergence of 0.001 eV/˚A.
  • Dataset construction: The C10H2/C10H+3 system was terminated when forces fell below 0.0015 eV/˚A.This force threshold is reported specifically for that system.
  • Dataset construction: NaCl and Au2-MgO datasets each contained two structurally different system types, with half of each dataset assigned to each case.Random trajectory sampling and additional Gaussian displacements were used to improve sampling near the structures of interest.
Loading 2009.06484v1…