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On representing chemical environments

Albert P. Bartók, Risi Kondor, Gábor Csányi

arXiv:1209.3140v3physics.comp-phcond-mat.mtrl-sci

TL;DR

Atomic-environment descriptors are crucial for comparing structures, identifying equivalent structures, and detecting similarities. The paper analyzes invariant descriptors and introduces SOAP, which defines similarity directly between atomic neighborhoods and remains connected to invariant descriptors. SOAP demonstrates excellent faithfulness for any neighbor count and produces more accurate, much more robust potential energy surfaces for small silicon clusters and bulk silicon.

  • Problem

    Atomic-environment descriptors are crucial for comparing structures, identifying equivalent structures, and detecting similarities.

    Source: §I · p. 1

  • Method

    The paper analyzes invariant descriptors and introduces SOAP, which defines similarity directly between atomic neighborhoods and remains connected to invariant descriptors.

    Source: §I · p. 2, §IV · p. 11, §VI · p. 17

  • Results

    SOAP demonstrates excellent faithfulness for any neighbor count and produces more accurate, much more robust potential energy surfaces for small silicon clusters and bulk silicon.

    Source: §VI · p. 18

  • Takeaways & Limitations

    The comparison shows that descriptor variants differ in faithfulness and convergence, while SOAP avoids the reported neighbor-count difficulties in the tested settings.

    Source: A. Reconstruction experiments · p. 15, §VI · p. 18

Abstract

from arXiv · show

We review some recently published methods to represent atomic neighbourhood environments, and analyse their relative merits in terms of their faithfulness and suitability for fitting potential energy surfaces. The crucial properties that such representations (sometimes called descriptors) must have are differentiability with respect to moving the atoms, and invariance to the basic symmetries of physics: rotation, reflection, translation, and permutation of atoms of the same species. We demonstrate that certain widely used descriptors that initially look quite different are specific cases of a general approach, in which a finite set of basis functions with increasing angular wave numbers are used to expand the atomic neighbourhood density function. Using the example system of small clusters, we quantitatively show that this expansion needs to be carried to higher and higher wave numbers as the number of neighbours increases in order to obtain a faithful representation, and that variants of the descriptors converge at very different rates. We also propose an altogether new approach, called Smooth Overlap of Atomic Positions (SOAP), that sidesteps these difficulties by directly defining the similarity between any two neighbourhood environments, and show that it is still closely connected to the invariant descriptors. We test the performance of the various representations by fitting models to the potential energy surface of small silicon clusters and the bulk crystal.

I. INTRODUCTION

Atomic-environment descriptors must preserve structural faithfulness while remaining invariant to physical symmetries and differentiable as atoms move. The paper frames descriptor choice around fitting potential energy surfaces, where a fixed-size representation is useful even as neighbor counts vary.

I. INTRODUCTION

  • A useful invariant representation must distinguish environments up to symmetry, while remaining continuous and differentiable as neighbor counts vary.For practical function fitting, the descriptor dimension should remain fixed even when the number of neighbors changes.Source: §I · pp. 1–2
  • The paper compares descriptor construction separately from the complex tasks of generating first-principles data and fitting potential energy surfaces.The introduction notes that no method had yet emerged as clearly optimal for representing atomic neighborhoods.Source: §I · p. 2
  • The paper links several established invariant descriptors: bond-order parameters are a subset of the bispectrum, and related angular components are polynomials of shared canonical invariants.The bispectrum provides an over-complete family that can be truncated to refine sensitivity.Source: §I · p. 2

II. POTENTIAL ENERGY SURFACE FITTING

  • Potential-energy models use local atomic descriptors inside a kernel that captures similarity between environments, making descriptor invariance and kernel choice central to model quality.The same kernel framework accommodates linear regression, neural-network-derived covariance functions, and squared-exponential kernels.Source: §II · pp. 2–3

III. DESCRIPTORS

The section surveys structural descriptors and their trade-offs in faithfulness, symmetry handling, differentiability, and scalability. Several representations capture some invariances but remain incomplete or unsuitable for fitting potential energy surfaces.

III. DESCRIPTORS

  • Weyl matrices encode rotation-, reflection-, and translation-invariant information, but permuting atoms reorders their rows and columns.The permutation-minimized reference metric compensates for this ordering dependence.Source: §III · p. 3
  • The reference distance between Weyl matrices minimizes mismatch over atom permutations, but is nondifferentiable when the minimizing permutation changes and intractable for large systems.It is used as a faithfulness benchmark for small systems, while later differentiable invariants are closely related to Weyl-matrix elements.Source: §III · p. 3
  • Eigenvalue descriptors are differentiable and permutation-invariant, but their N values fall short of the 3N − 3 configuration dimensions and continuity with changing neighbor counts is unclear.A similar descriptor was used to fit molecular atomization energies.Source: §III · p. 3
  • Pairwise-distance histograms are incomplete: two distinct tetrahedra can share the same six distances, while alignment-based distances introduce cusps through reordering and rotation/reflection minimization.The alignment distance is complete and symmetry-invariant, but those cusps make it unsuitable for parameterizing potential energy surfaces.Source: §III · p. 4

A. Bond-order parameters

Bond-order parameters are built from an atomic neighbor density expanded in spherical harmonics, producing rotationally invariant measures of local and global order. The construction initially focuses on angular information, with radial information added later.

A. Bond-order parameters

  • The atomic neighbor density sums neighbor contributions within a cutoff, with species-specific weights and optional binary or smooth distance cutoffs.The density is already invariant to permutations of neighbors because their order only changes the summation order.Source: A. Bond-order parameters · p. 4
  • Expanding the angular density in spherical harmonics yields coefficients used to construct rotationally invariant bond-order parameters.The paper distinguishes atomic order parameters from global parameters averaged over atoms in the system.Source: A. Bond-order parameters · p. 4
  • Second- and third-order bond-order parameters characterize local structure; their values vary with crystalline symmetry and global parameters vanish in disordered phases such as liquids.Radial information is omitted in this derivation and is incorporated later.Source: A. Bond-order parameters · pp. 4–5

B. The power spectrum

The power spectrum and bispectrum construct rotational invariants by combining spherical-harmonic expansion coefficients. The bispectrum extends the power spectrum through angular-channel coupling and contains the power spectrum and bond-order parameters as special cases.

B. The power spectrum

  • The power spectrum is rotationally invariant and is related by a simple scaling to the second-order bond-order parameters.Its invariance follows from the unitary transformation of spherical-harmonic coefficients under rotation.Source: B. The power spectrum · pp. 5–6
  • Although the power spectrum is incomplete for general functions on the sphere, its components are polynomials of Weyl invariants for atomic neighbor densities and may be over-complete at fixed neighbor count.The paper supports this qualification for densities formed from a limited number of delta functions.Source: B. The power spectrum · p. 6

C. The bispectrum

  • The bispectrum couples angular-momentum channels to produce rotationally invariant scalars, with reflection-odd components removed or made absolute when reflection invariance is required.Its diagonal elements include the third-order bond-order parameters, and particular components reproduce power-spectrum terms.Source: C. The bispectrum · pp. 6–7
  • The paper presents the power spectrum and bispectrum as angular descriptors built from spherical-harmonic coefficients and their coupled products.For an example with three neighbors, the illustrated terms depend on angles between neighbor bonds.Source: C. The bispectrum · pp. 6–7

D. Radial basis

Radial information can be added to the spherical-harmonic representation with radial basis functions, but weak overlap can leave atoms in different radial shells poorly coupled. Broadly sensitive or explicitly cross-coupled bases address this issue with trade-offs in sensitivity or descriptor count.

D. Radial basis

  • Radial basis functions add distance information to the spherical-harmonic expansion, which otherwise neglects neighbours’ distances from the central atom.For a non-orthonormal basis, the coefficients can be transformed using the overlap matrix; either the primed or transformed coefficients can be used to construct the bispectrum.Source: D. Radial basis · p. 7
  • The radial basis functions are combined with spherical harmonics to form invariants describing three-dimensional atomic neighbourhoods.Using separate radial channels without sufficient overlap can make the resulting representation poor.Source: D. Radial basis · pp. 7–8
  • Weakly overlapping radial functions can leave different shells weakly coupled, creating spurious quasi-invariance to rotating subsets of atoms at similar distances.Avoiding this requires basis functions sensitive across a wide distance range, though that can reduce each function’s sensitivity and offers no guaranteed satisfactory choice for every system.Source: D. Radial basis · p. 8
  • The proposed polynomial radial functions return smoothly to zero at the cutoff with their first and second derivatives also zero.The functions are constructed from cubic and higher-order polynomials and normalized over the interval from zero to the cutoff.Source: D. Radial basis · p. 8
  • Coupling different radial channels prevents decoupling, but increases the number of invariants quadratically or cubically with the number of radial basis functions.Rotational invariance must apply globally, so angular momentum channels associated with different radial functions need to be coupled.Source: D. Radial basis · pp. 7–8

E. 4-dimensional power spectrum and bispectrum

The four-dimensional construction maps the atomic neighbourhood density onto a hypersphere and expands it in hyperspherical harmonics. Its invariants combine radial and angular information while retaining the required symmetry properties.

E. 4-dimensional power spectrum and bispectrum

  • The method projects the neighbourhood density within a cutoff onto the surface of a four-dimensional sphere, avoiding an explicitly introduced radial basis set.The projection uses a parameter r0 greater than the cutoff radius.Source: E. 4-dimensional power spectrum and… · pp. 8–9
  • The projection parameter affects radial sensitivity: r0 at the cutoff loses angular information at that boundary, while an excessively large r0 requires many basis functions.A broad range works in practice; the authors used r0 = 4/3 rcut.Source: E. 4-dimensional power spectrum and… · p. 9
  • Hyperspherical harmonics provide an orthonormal basis on the sphere, and expansion coefficients are calculated using the inner product defined there.The harmonics correspond to components of rotational matrices, with j taking half-integer values.Source: E. 4-dimensional power spectrum and… · p. 9
  • The construction anchors the neighbourhood by adding the central atom, removing unphysical invariance along the radial angle and yielding invariants with radial and angular information.The resulting invariants have the required symmetry properties, with r0 as their only adjustable parameter in the definition.Source: E. 4-dimensional power spectrum and… · p. 10
  • The truncated representation omits reflection-changing components and keeps only diagonal bispectrum elements to reduce redundancy.The number of full and diagonal bispectrum components depends on the band limit jmax.Source: E. 4-dimensional power spectrum and… · p. 10, Table II · p. 10

F. Parrinello-Behler descriptor

The Parrinello-Behler descriptor uses two- and three-body symmetry functions, with parameters that can generate an arbitrary number of invariants.

F. Parrinello-Behler descriptor

G. Angular Fourier Series

The Angular Fourier Series descriptor uses orthogonal polynomials of angular invariants, while radial information is incorporated using radial basis functions.

G. Angular Fourier Series

  • The angular parts of the power spectrum, bispectrum, and Parrinello-Behler descriptors are polynomials of the canonical invariants Pii′ cos(mθii′).Source: G. Angular Fourier Series · p. 10
  • The Angular Fourier Series descriptor uses Chebyshev polynomials Tl as an orthogonal polynomial system of the basic invariants.Source: G. Angular Fourier Series · p. 10
  • Radial information is incorporated into the AFS descriptor using the radial basis functions defined earlier.Source: G. Angular Fourier Series · p. 10
  • The illustrated AFS angular basis functions use lmax = 4.Source: Figure 6 · p. 11

IV. A SIMILARITY MEASURE BETWEEN ATOMIC ENVIRONMENTS

SOAP compares environments through a similarity kernel built from their neighbour densities, rather than requiring a descriptor first. Rotational averaging and Gaussian smoothing give a symmetry-invariant, smooth comparison, with the kernel’s sensitivity adjustable by an exponent.

IV. A SIMILARITY MEASURE BETWEEN ATOMIC ENVIRONMENTS

  • A good similarity measure must be symmetry-invariant, well-defined for identical or very different environments, and smooth with respect to Cartesian coordinates.The paper defines similarity directly from the inner product of the two neighbour densities.Source: §IV · p. 11
  • The overlap of two neighbour densities is permutation-invariant; integrating it over rotations produces a rotationally invariant similarity kernel.The kernel retains angular information when n ≥ 2, but loses it for n = 1.Source: §IV · p. 11
  • For n = 1, the kernel loses angular information, whereas for n ≥ 2 it retains the environments’ angular information.The construction also requires evaluating an angular integral.Source: §IV · p. 11

A. Analytic evaluation a smooth similarity kernel

  • Using Dirac-delta neighbour densities would make the similarity discontinuous for environments with close but nonidentical atomic positions, so the method instead uses Gaussians.The Gaussian density is expanded in spherical harmonics.Source: A. Analytic evaluation a smooth… · p. 11
  • Raising the overlap kernel to a positive integer power preserves positive definiteness, and powers ζ ≥ 2 accentuate sensitivity to atomic-position changes.The general SOAP kernel normalizes the powered overlap by the self-overlaps of both environments.Source: A. Analytic evaluation a smooth… · p. 12

B. Radial basis and relation to spectra

Expanding neighbour densities in radial basis functions connects SOAP kernels to invariant spectral descriptors. The paper relates the kernel to power spectra and bispectra and illustrates its convergence as angular and radial basis sizes increase.

B. Radial basis and relation to spectra

  • The radial-basis expansion expresses neighbour densities with coefficients and links the SOAP kernel to power-spectrum and bispectrum representations.Computing the coefficients for neighbour pairs can otherwise become expensive.Source: B. Radial basis and relation to spectra · p. 12
  • SOAP is equivalent to using SO(3) power-spectrum or bispectrum descriptors with Gaussian neighbour densities and a dot-product covariance kernel.With a radial basis, the kernel in equation (39) is the dot product of the environments’ power spectra.Source: B. Radial basis and relation to spectra · p. 12
  • SOAP removes some earlier ad hoc choices while α controls smoothness and ζ controls sensitivity to changes in atomic positions.These parameters tune the similarity kernel used in the spectral representations.Source: B. Radial basis and relation to spectra · pp. 12–13
  • Figure 7 examines kernel convergence for environments with 15 neighbours: its top panel varies spherical harmonics, while its bottom varies radial basis functions at lmax = 16.The caption reports α = 0.4, ζ = 1.0, and a converged kernel value of −0.842735.Source: B. Radial basis and relation to spectra · p. 12, Figure 7 · p. 13

V. NUMERICAL RESULTS

Reconstruction tests show that truncated descriptors lose faithfulness as neighbourhoods grow, while SOAP reconstructs all tested structures accurately. Cutoff sensitivity and angular truncation also affect reconstruction quality.

A. Reconstruction experiments

  • With the tested parameter set, larger randomisations expose poor representation by all descriptors for clusters with n > 10.The reference distance increases with neighbour count; at n > 10 the neighbour configuration becomes impossible to determine from the descriptor.Source: Figure 9 · p. 14, A. Reconstruction experiments · p. 15
  • At a 9 Å cutoff, faithful reconstruction extends to n ≤ 9 for the SO(3) power spectrum and AFS, and n ≤ 12 for the SO(4) bispectrum.The peak near n = 8 seen at the 6 Å cutoff is absent at 9 Å.Source: A. Reconstruction experiments · p. 15, Figure 11 · p. 15
  • AFS reconstruction improves monotonically with higher angular resolution, but its increasingly oscillatory basis functions may suit smooth PES fitting less well.Figure 12 compares different AFS angular truncations.Source: A. Reconstruction experiments · p. 15
  • SOAP with lmax = 6 reconstructs all tested structures without degradation as neighbour count increases.The comparison varied angular truncation for SOAP and AFS.Source: A. Reconstruction experiments · p. 15
  • Descriptor matching reaches zero objective for complete or over-complete descriptors and SOAP, while incomplete descriptors retain a non-zero reference distance.This check addresses whether reconstruction failures reflect minimisation getting stuck.Source: A. Reconstruction experiments · p. 15

B. Gaussian Approximation Potentials

The GAP tests compare descriptors on silicon clusters and bulk silicon. SOAP outperforms the other tested descriptors for cluster potentials and yields more robust bulk elastic constants, while high angular resolution can degrade bispectrum fits.

B. Gaussian Approximation Potentials

VI. CONCLUSION

  • The conclusion is that SOAP avoids the truncated-descriptor faithfulness difficulty and gives more accurate, robust potential energy surfaces for the tested silicon clusters and bulk crystal.The paper reports excellent faithfulness for any number of neighbours.Source: §VI · pp. 17–18

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