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Efficient Approximations of Complete Interatomic Potentials for Crystal Property Prediction

Yuchao Lin, Keqiang Yan, Youzhi Luo, Yi Liu, Xiaoning Qian, Shuiwang Ji

arXiv:2306.10045v9physics.chem-phcs.LG

TL;DR

Crystal GNNs that connect only nearby atoms do not explicitly capture distant interactions across a crystal’s infinite periodic repetitions. PotNet incorporates interatomic potentials and approximations of their complete sums into message passing; it reports consistent improvements, including best performance on all five JARVIS tasks.

  • Problem

    Distance-threshold crystal graphs omit explicit interactions between distant atoms, despite the infinite periodic repetition of unit cells.

    Source: §1 · p. 2

  • Method

    PotNet uses Coulomb, London dispersion, and Pauli repulsion potentials and approximates complete infinite potential sums for integration into GNN message passing.

    Source: §1 · p. 2, §3.3 · p. 5

  • Results

    PotNet achieves the best performance on all five JARVIS tasks, while interatomic and complete interatomic potentials yield consistent performance improvements with reasonable computational costs.

    Source: §1 · p. 2, §4.2 · p. 8

  • Takeaways & Limitations

    The work presents a theoretically principled and practically effective framework for crystal modeling using complete potentials.

    Source: §6 · p. 9

    The interatomic potential formulation does not account for interactions involving more than two atoms, and explicit angular and many-body interactions may improve precision.

    Source: §5 · p. 9

Abstract

from arXiv · show

We study property prediction for crystal materials. A crystal structure consists of a minimal unit cell that is repeated infinitely in 3D space. How to accurately represent such repetitive structures in machine learning models remains unresolved. Current methods construct graphs by establishing edges only between nearby nodes, thereby failing to faithfully capture infinite repeating patterns and distant interatomic interactions. In this work, we propose several innovations to overcome these limitations. First, we propose to model physics-principled interatomic potentials directly instead of only using distances as in many existing methods. These potentials include the Coulomb potential, London dispersion potential, and Pauli repulsion potential. Second, we model the complete set of potentials among all atoms, instead of only between nearby atoms as in existing methods. This is enabled by our approximations of infinite potential summations, where we extend the Ewald summation for several potential series approximations with provable error bounds. Finally, we propose to incorporate our computations of complete interatomic potentials into message passing neural networks for representation learning. We perform experiments on the JARVIS and Materials Project benchmarks for evaluation. Results show that the use of interatomic potentials and complete interatomic potentials leads to consistent performance improvements with reasonable computational costs. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS/tree/main/OpenMat/PotNet).

1. Introduction

Crystal GNNs face the challenge of representing interactions across an infinitely repeated unit cell, while nearby-atom graph edges omit distant interactions. PotNet uses interatomic potentials and approximations to model the complete set of interactions in message-passing networks.

1. Introduction

  • Molecular GNN research expanded from 2D atom-and-bond graphs to 3D models because molecular property prediction may require geometry that 2D representations omit.The 3D methods use atom coordinates and geometric information such as distances and angles during message passing.Source: §1 · p. 1
  • Because a crystal unit cell repeats in three dimensions, capturing infinite-range interatomic interactions is a key modeling challenge; distance-threshold graphs omit distant interactions explicitly.The paper treats infinite repetition as an approximately accurate model of crystals at the atomic level.Source: §1 · pp. 1–2
  • PotNet uses Coulomb, London dispersion, and Pauli repulsion potentials as edge features and approximates their complete infinite sums for message passing.Experiments on JARVIS and Materials Project report consistent performance improvements with reasonable computational costs.Source: §1 · p. 2

2. Background and Related Work

The background defines crystals as periodic unit-cell structures and frames property prediction around their energy-related properties. It reviews interatomic-potential approaches and the development of deep learning models for crystal prediction.

2.1. Crystal Representation and Property Prediction

  • A crystal is represented by a unit cell repeated periodically in 3D, and the prediction objective includes total energy and other energy-related properties.The unit cell contains the smallest repeatable structure; the lattice matrix describes its repetitions.Source: §2.1 · p. 2

2.2. Crystal Property Prediction with Interatomic Potentials

  • Classical crystal-energy methods approximate total energy by summing interatomic potentials, including Coulomb, London dispersion, and Pauli repulsion interactions.The passages link Coulomb interactions to charged atoms, London dispersion to Van der Waals interactions, and Pauli repulsion to electron-orbit exclusion.Source: §2.2 · p. 2

2.3. Crystal Property Prediction with Deep Learning

  • Physics-based methods can be crystal-type-specific, while Coulomb matrices are chiefly used for ionic and metallic materials and lack permutation invariance.The passage also describes chemical-formula sequence models for crystal property prediction.Source: §2.3 · pp. 2–3
  • Crystal deep learning progressed from CGCNN radius graphs to varied 3D GNNs, with Matformer described as a leading architecture using periodic invariance and patterns.These models represent crystals as 3D graphs for property prediction.Source: §2.3 · p. 3

3. Method

PotNet represents crystals using physics-principled interatomic potentials and complete interactions, incorporating their summations into GNN message passing. Its method uses Ewald-based approximations to compute infinite potential sums efficiently with provable error bounds.

3.1. Approximating Crystal Energy with Complete Interatomic Potentials

  • The embedded-atom formulation approximates crystal energy with pairwise interatomic potentials and a learned function of local electron density.Direct evaluation of the function over complete potential sums is computationally expensive, motivating neural-network approximation.Source: §3.1 · p. 3

3.2. Limitations of Existing Deep Learning Methods

  • PotNet addresses radius-graph limits by using potentials directly and including interactions across the infinite crystal, whose algebraically decaying contributions accumulate.For London dispersion with p=6, a radius of 56 is required to approximate the sum with 0.01 absolute error; common radius graphs average about r=1.Source: §3.2 · p. 4

3.3. Message Passing with Complete Interatomic Potentials

  • PotNet uses interatomic potentials as edge features and aggregates the infinite interactions between each pair of unit-cell atoms into one edge for GNN message passing.The edge potential sum includes interactions from an atom and all its periodic repetitions.Source: §3.3 · pp. 4–6

3.4. Efficient Computation of Infinite Potential Summation

  • Ewald summation splits infinite potential sums into a real-space part and a Fourier-space part, yielding convergent approximations with provable error bounds.The method represents both parts using incomplete Bessel-function sums, whose approximation error is bounded by the Gaussian Lattice Sum.Source: §3.4 · pp. 6–7
  • The Ewald-based approach directly approximates London dispersion and Pauli-potential sums, while computing Coulomb sums using established mathematics and analytic continuation.The method also supports Lennard-Jones, Morse, and screened Coulomb potential sums; the authors identify its Pauli-potential summation as a first use of incomplete Bessel functions.Source: §3.4 · p. 7

4. Experimental Studies

PotNet outperforms strong baselines across the Materials Project and JARVIS benchmarks, with ablations reporting gains from potential features and infinite summation. The experiments also report training and prediction-time comparisons.

4.1. Experimental Setup

  • PotNet combines local radius graphs using Coulomb-potential edge features with infinite crystal graphs using Coulomb, London dispersion, and Pauli potentials.The experiments use Materials Project and JARVIS benchmarks; periodic-table information and transformer operations are discussed only in the supplementary section, not the main paper.Source: §4.1 · pp. 7–8

4.2. Experimental Results

  • PotNet is four times faster than ALIGNN in total training and inference, and faster than Matformer by 34% per epoch and 47% in inference.These comparisons use JARVIS formation-energy prediction.Source: Table 3 · p. 8, §4.2 · p. 8
  • PotNet consistently outperforms other state-of-the-art methods on four Materials Project tasks, including properties with only 4,664 training samples.The benchmark contains 69,239 crystals; evaluated properties are formation energy, band gap, bulk moduli, and shear moduli.Source: §4.2 · p. 8
  • PotNet achieves the best performance on all five JARVIS property-prediction tasks using the same splits and retrained baselines as Matformer.The tasks are formation energy, bandgap (OPT), bandgap (MBJ), total energy, and Ehull.Source: §4.2 · p. 8
  • PotNet’s mean prediction time per crystal is at the millisecond level, faster than ALIGNN but slightly slower than Matformer.Prediction time includes preprocessing and model inference; the comparison is with the methods’ JARVIS results.Source: Table 4 · p. 8, §4.2 · p. 8

4.3. Ablation Studies

  • With the architecture held fixed, replacing Euclidean distances with Coulomb potentials improves the base model from 0.0363 to 0.0301.The compared variants are Base + Euclidean and Base + Potential.Source: §4.3 · p. 9
  • Adding infinite potential summation further improves formation-energy prediction from 0.0301 to 0.0294.The comparison is between Base + Potential and the full PotNet model with infinite summation.Source: §4.3 · p. 9

5. Limitation

The modeled interatomic potentials do not account for interactions involving more than two atoms. The authors suggest incorporating angular and many-body interactions as a possible route to more precise simulations.

5. Limitation

  • The current potentials are limited to pairwise interactions; explicitly modeling angular and many-body interactions may improve simulation precision.The authors propose extending the approach to three-body potentials and infinite summation.Source: §5 · p. 9

6. Conclusion

This section describes PotNet’s use of interatomic potentials and efficient approximations to complete potential sums for crystal-property modeling, supported by mathematical constructions and convergence arguments. It also presents analytic continuation for approximating total contributions when individual sums cancel, with numerical examples assessing approximation results, error bounds, actual errors, and runtimes.

6. Conclusion

  • PotNet combines interatomic potentials with efficient approximations to their complete set among atoms, and complete potentials yield consistent performance improvements.The authors identify improving approximation error bounds and summation algorithms as future work.Source: §6 · p. 9

A. Gaussian Lattice Sum

  • The Gaussian lattice sum decays rapidly, can be expressed using the Riemann theta function, and converges under the stated positive-definiteness condition.The condition is that the imaginary part of the theta-function parameter is positive definite.Source: A. Gaussian Lattice Sum · p. 14

B.1. Fast Approximation of the Incomplete Bessel Function

  • The incomplete Bessel function admits an O(n) approximation, with analytic continuation used for real ν and a Taylor approximation available when ν = 0 and x, y are small.The cited Taylor expansion has an error bound described in Nestler et al. (2015).Source: §B.1 · pp. 14–15, B. Incomplete Bessel Function · p. 14

B.2. Convergence of Incomplete Bessel Function Summation

  • The incomplete Bessel-function lattice summation converges, and truncating it to a finite ellipsoid leaves an error bounded by the Gaussian lattice sum.The bound applies for a prefixed R satisfying the stated conditions.Source: §B.2 · p. 16

C.1. Integral Transformation

  • Integral transformations express the lattice sums for inverse-power and exponential potentials in integral form.The inverse-power case uses a Mellin transform, while the exponential case uses an inverse Laplace transform.Source: §C.1 · p. 17

C.2. Calculating Integral Summation

  • Both direct-space and Fourier-space potential sums reduce to incomplete Bessel-function summations, so the potential sum converges and can be approximated.The Fourier-space representation is derived using Poisson summation on the reciprocal lattice.Source: §C.2 · pp. 18, 20

C.3. Analytic Continuation of Potential Summations

  • Analytic continuation extends the potential-summation domain, enabling Coulomb summation for the crystal setting p = 1 and d = 3.The continuation excludes the poles at C = 0 and C = d/2.Source: §C.3 · p. 20

C.4. Generalized Epstein Zeta Function and Analytic Continuation

  • The potential sum is a special case of the generalized Epstein zeta function, whose analytic continuation supports summing Coulomb potentials for ionic crystals, including the NaCl Madelung constant.The function can also be formed as a linear combination of individual potential sums, which is useful when the phase vector is learned.Source: §C.4 · pp. 20, 22
  • Analytic continuation of the zeta function evaluates conditionally convergent series and supports approximating total potential contributions when individual potential summations cancel.The paper also demonstrates analytic continuation for calculating the Madelung constant of NaCl in Appendix E.Source: §C.4 · p. 22
  • The numerical examples compare approximations across grid lengths, reporting estimated error upper bounds, truth errors, and computation times.Implementation details are in Appendix C.5.Source: Table 6 · p. 23, §C.4 · p. 23

C.5. Implementation and Numerical Examples of Approximation

The approximation implementation selects lattice-grid points within ellipsoidal bounds to evaluate direct and Fourier summation terms, with estimated error bounds. The method also extends to additional interatomic potentials, with example computations reported at millisecond scale.

C.5. Implementation and Numerical Examples of Approximation

  • The algorithm selects grid points within ellipsoidal bounds to evaluate direct and Fourier summation terms, estimating R and an error bound when R is unknown.The grid length is 2r, with r > R; the direct and Fourier procedures use separate selection conditions.Source: §C.5 · pp. 23–24
  • The implementation reports millisecond-scale evaluation times for numerical examples of the approximation.The examples use Cython, the GNU Scientific Library, and ScaFaCoS.Source: §C.5 · p. 24

C.6. Potential Summation Extensions

  • The summation method extends to Lennard-Jones, Morse, and screened Coulomb potentials.Lennard-Jones and Morse summations are reduced to pairs of summations already handled by the method; screened Coulomb is fitted using a Laplace transform.Source: §C.6 · pp. 24–25

D.1. PotNet Implementation

PotNet combines atomic features with local Coulomb potentials and global summations of Coulomb, London dispersion, and Pauli potentials. Its input and interaction blocks embed these edge features for message passing, followed by graph-level pooling and scalar prediction.

D.1. PotNet Implementation

  • PotNet uses atomic features alongside local Coulomb potentials and global summations of Coulomb, London dispersion, and Pauli potentials.The model combines the three infinite-potential features into a single feature for the network.Source: §D.1 · pp. 25–26
  • The input block maps atomic features to 256 dimensions and embeds potential features using radial basis function kernels.Coulomb potentials use 256 kernels; infinite-potential summations use 64 kernels followed by an MLP up-projection to 256 dimensions.Source: §D.1 · pp. 25–26
  • Each interaction layer updates node features using neighboring-node features and concatenated embeddings of the connecting edge potentials.The interaction pattern follows CGCNN.Source: §D.1 · p. 26
  • The readout averages node features and maps the aggregated 256-dimensional vector to a scalar output.The readout uses an AvgPooling layer followed by a Linear layer.Source: §D.1 · p. 26

D.2. PotNet Improvement

Among the reported variants, PotNet-T performs better than PotNet and PotNet-C, while PotNet-C is comparable to PotNet. PotNet-T has lower computational efficiency despite only a modest performance increase.

D.2. PotNet Improvement

  • PotNet-T outperforms PotNet and PotNet-C, while PotNet-C performs comparably to PotNet.The comparison is reported for JARVIS formation energy; Table 7 also reports training time.Source: Table 7 · p. 26, §D.2 · p. 26
  • PotNet-C adds periodic-table element-class information, while PotNet-T adds a transformer operation on infinite-potential features.The transformer output is added to the local-graph node output; the combined structure with periodic-table information is called PotNet-T.Source: §D.2 · p. 26
  • PotNet-T has lower computational efficiency despite its modest performance increase.This trade-off qualifies the performance advantage over PotNet and PotNet-C.Source: §D.2 · pp. 26–27

D.3. Dataset Information

The dataset comparison reports dataset size and unit-cell statistics. Typical 4 Å or 8 Å cutoffs span roughly one to eight unit-cell neighbor hops.

  • Table 8 compares the Materials Project and JARVIS datasets by size, average atoms per cell, mean cell lengths, and minimum cell length.Source: Table 8 · p. 27
  • Minimum cell lengths range approximately from 1 Å to 2 Å.Source: §D.3 · p. 27
  • Common 4 Å or 8 Å cutoffs cover, on average, only one-hop neighbors and at most eight-hop neighbors.Source: §D.3 · p. 27

D.4. Full Ablation on JARVIS Dataset

On JARVIS, adding infinite potential summation features improves performance across all evaluated metrics, with particularly notable gains for BandGap (MBJ) and Ehull.

  • Infinite potential summation features improve performance across all metrics evaluated in the JARVIS ablation.Source: §D.4 · p. 27
  • The ablation study evaluates the contribution of infinite potential summations on the JARVIS dataset.Source: §D.4 · p. 27
  • BandGap (MBJ) and Ehull show particularly substantial predictive-accuracy enhancements.Source: §D.4 · p. 27

D.5. Cutoff Experiments

Cutoff experiments compare established GNNs and a PotNet variant on JARVIS formation energy. Larger cutoffs correlate with better performance, but create practical training and preprocessing constraints.

  • The tested approaches were chosen for wide use and because they require no graph construction beyond radius crystal graphs.Source: §D.5 · p. 27
  • The comparison includes SchNet, GATGNN, and PotNet without infinite potential summation or local Coulomb potential, capping each atom at 16 neighbors.Source: §D.5 · p. 27
  • Increasing the cutoff correlates with improved formation-energy performance across the three tested methodologies, but SchNet has no final result at 50 Å after gradient explosion.Source: §D.5 · p. 27
  • As cutoff values increase, preprocessing time for all 55,722 JARVIS crystals escalates to an unmanageable extent.Source: §D.5 · p. 27, Table 10 · p. 27

E. Linear Energy Modeling using Infinite Potential Summation

The section illustrates how infinite potential summations approximate energies for ionic crystals, where Coulomb interactions dominate. For NaCl, adding a repulsion term yields a Born–Landé estimate, while the network can learn summation coefficients and nonlinearity.

  • For ionic NaCl, MgO, and LiF, Coulomb interactions dominate, so their total electrostatic energies are considered through Coulomb potential summations.Source: E. Linear Energy Modeling using… · pp. 28–29
  • NaCl's symmetry reduces the calculation to atoms a, b, c, and d, using their infinite Coulomb interactions to approximate the energy of Na+.Source: E. Linear Energy Modeling using… · p. 28
  • The NaCl Coulomb summation yields the Madelung constant −1.7475646 and an electrostatic energy approximation of 8.81 eV; with Born exponent n = 9, the total-energy estimate is 7.84 eV.Source: E. Linear Energy Modeling using… · p. 28
  • The approximations are rough relative to ground-truth energies, while the network learns summation coefficients and supplies nonlinearity.The coefficients depend on the system's charge distribution, which is unknown initially.Source: E. Linear Energy Modeling using… · p. 29
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