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Gaussian Approximation Potentials: the accuracy of quantum mechanics, without the electrons
Albert P. Bartók, Mike C. Payne, Risi Kondor, Gábor Csányi
TL;DR
Analytic interatomic potentials can be limited in accuracy, while explicit-electron methods are computationally restricted. This paper builds finite-range potentials by interpolating quantum-mechanical reference data with symmetry-aware neighborhood representations and regression on energies and forces. The resulting models achieve low energy errors and agree with experiment in high-temperature tests.
Problem
Explicit-electron models are limited in practice to a few thousand atoms, while simple analytic interatomic potentials have limited accuracy.
Source: p. 1Method
The method interpolates quantum-mechanical reference results to model atomic energies from neighborhood similarities, using energies and forces with sparsification.
Source: pp. 2–3, §2.2 · p. 8Results
For all three materials, RMS energy errors are less than 1 meV/at; GAP also agrees well with experiment at high temperatures.
Source: pp. 3–4Takeaways & Limitations
The framework produces finite-range potentials that match ab initio potential-energy surfaces at a fraction of the cost and can describe reactive carbon transitions.
Source: p. 4Atomic energies are not uniquely determined from electronic-structure calculations; the available total energies and forces are linear combinations of atomic contributions.
Source: §2.2 · pp. 7–8
Abstract
from arXiv · showhide
We introduce a class of interatomic potential models that can be automatically generated from data consisting of the energies and forces experienced by atoms, derived from quantum mechanical calculations. The resulting model does not have a fixed functional form and hence is capable of modeling complex potential energy landscapes. It is systematically improvable with more data. We apply the method to bulk carbon, silicon and germanium and test it by calculating properties of the crystals at high temperatures. Using the interatomic potential to generate the long molecular dynamics trajectories required for such calculations saves orders of magnitude in computational cost.
1 Bispectrum
The bispectrum represents an atomic neighborhood using coefficients of a density expanded in hyperspherical harmonics. Combining those coefficients yields rotation- and permutation-invariant features, which are truncated to set spatial resolution.
1 Bispectrum
- Hyperspherical harmonics provide an orthonormal basis for representing density on the surface of a 4D sphere.The expansion coefficients are calculated by inner products, and coefficient vectors transform under unitary operations such as rotations.Source: §1 · p. 6
- The direct product of rotational matrices is decomposed through a unitary transformation using a 4D analogue of Clebsch-Gordan coefficients.This construction yields coefficient combinations that transform as rotational matrices.Source: §1 · p. 6
- The cubic rotational invariants, or bispectrum, are constructed from these transformed coefficients and are invariant to rotations and atom permutations.The resulting three-index array is truncated at Jmax, limiting the spatial resolution of the neighborhood description.Source: §1 · p. 6
- The truncated bispectrum contains 4, 23, and 69 elements for Jmax values of 1, 3, and 5, respectively.Source: §1 · p. 6
2 Gaussian Process Regression
Gaussian process regression uses a covariance function over bispectrum representations to predict atomic energies from reference data. It accommodates observed total energies and forces as linear combinations of unknown atomic quantities, while sparsification reduces the data representation.
2 Gaussian Process Regression
- The regression setup defines raw and sparse neighborhood configurations, their bispectra, and covariance matrices linking raw, sparse, and test configurations.A pseudo-covariance matrix is formed for sparse configurations.Source: §2 · p. 7
2.1 Covariance function (kernel)
- A Gaussian kernel assigns a separate spatial-scale hyperparameter to each feature, allowing a more flexible covariance than a single-scale Gaussian kernel.The covariance equations cover function values and derivatives at either or both configurations.Source: §2.1 · p. 7
2.2 Linear combinations
- Total energies and forces provide linear combinations of atomic energies and their derivatives, rather than uniquely observed atomic contributions.The regression represents these observations through a matrix relating them to the unobserved atomic quantities.Source: §2.2 · pp. 7–8
2.3 Putting it all together
- Sparsification and linear-combination observations are combined to express the atomic energy without requiring the unobserved atomic values.The method selects a predetermined number of sparse configurations, much smaller than the full data set.Source: §2.3 · p. 8
3 The data: density functional theory
The reference data were generated with density functional theory using material-specific approximations and computational settings.
3 The data: density functional theory
- Carbon reference data used the local density approximation, while silicon, germanium, GaN, and iron used the PBE generalized gradient approximation.Source: §3 · p. 8
- For DFT data, electronic Brillouin zones were sampled using Monkhorst-Pack k-point grids, with spacing at most 0.3 Å^-1 for insulators; the passage also reports spacing at most 0.14 Å^-1, without specifying its material condition.Source: §3 · p. 8
- Plane-wave cutoffs were 350, 300, 300, 350, and 500 eV for carbon, silicon, germanium, GaN, and iron, respectively.Ultrasoft pseudopotentials used material-specific valence-electron counts, and energies were extrapolated to correct for finite basis sets.Source: §3 · p. 8
4 Testing the GAP parameters
For diamond, force-error distributions improve as the cutoff radius increases, and they also improve as bispectrum spatial resolution increases.
4 Testing the GAP parameters
- Increasing the cutoff radius improves the distribution of GAP force errors for diamond.Increasing Jmax, which raises bispectrum spatial resolution, also improves the distribution.Source: §4 · p. 8
5 Potential for gallium nitride
For GaN, the GAP model adds a fixed-charge Ewald sum to its local energy to represent long-range Coulomb interactions. Its phonon spectrum shows LO/TO splitting, indicating that the model captures the long-range ionic interactions correctly.
5 Potential for gallium nitride
- The GaN model augments local GAP energy with an Ewald sum of fixed charges (+1 for Ga and -1 for N).DFT energies and forces used in regression had this Coulomb contribution subtracted.Source: §5 · p. 8
- Figure 3 presents force errors for the Ewald-augmented GaN GAP model relative to DFT forces.The supplied passages identify the comparison but do not report specific error values.Source: §5 · p. 8, Figure 3 · p. 10
- LO/TO splitting in the GaN phonon spectrum shows that the model captures long-range ionic interactions correctly.The spectrum compares GAP (solid lines) with PBE-DFT (open squares).Source: §5 · p. 8, Figure 4 · p. 10