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A fingerprint based metric for measuring similarities of crystalline structures
Li Zhu, Maximilian Amsler, Tobias Fuhrer, Bastian Schaefer, Somayeh Faraji, Samare Rostami, S. Alireza Ghasemi, Ali Sadeghi, Migle Grauzinyte, Christopher Wolverton, Stefan Goedecker
TL;DR
Comparing crystalline structures matters for analyzing structure collections, but cell vectors and atomic coordinates are unsuitable fingerprints because their representations are non-unique and noise-sensitive. The paper introduces local-environment fingerprints and matches atoms to define configurational distances; the authors report that these distances satisfy metric properties and distinguish structures in their tests.
Problem
Structure collections need reliable configurational distances, but cell-and-coordinate representations are non-unique and can change discontinuously with noise.
Source: §I · p. 1Method
The method characterizes each atom by a local-environment fingerprint and defines crystal distances through matching atomic fingerprints.
Source: §I · p. 2, §II · p. 2Results
The authors report that their fingerprint defines metric configurational distances and distinguishes structures, including cases where global Oganov and BCM fingerprints classify defect structures as identical.
Source: §VI · p. 6, §V · p. 6Takeaways & Limitations
The fingerprint provides a way to measure crystalline-structure similarity or dissimilarity for distinguishing structures.
Source: §I · p. 2, §VI · p. 6
Abstract
from arXiv · showhide
Measuring similarities/dissimilarities between atomic structures is important for the exploration of potential energy landscapes. However, the cell vectors together with the coordinates of the atoms, which are generally used to describe periodic systems, are quantities not suitable as fingerprints to distinguish structures. Based on a characterization of the local environment of all atoms in a cell we introduce crystal fingerprints that can be calculated easily and allow to define configurational distances between crystalline structures that satisfy the mathematical properties of a metric. This distance between two configurations is a measure of their similarity/dissimilarity and it allows in particular to distinguish structures. The new method is an useful tool within various energy landscape exploration schemes, such as minima hopping, random search, swarm intelligence algorithms and high-throughput screenings.
I. INTRODUCTION
Crystalline-structure collections need reliable metric distances to identify duplicates, characterize structures, and maintain diversity. Cell-and-coordinate representations are non-unique and can respond discontinuously to noise, motivating fingerprints based on atomic environments and atom matching.
I. INTRODUCTION
- Metric distances are needed to identify duplicates and insufficiently characterized structures, compare noisy experimental or theoretical structures, and maintain diversity in structure prediction.Experimental measurements and incompletely converged calculations can leave nominally identical structures with nonzero differences.Source: §I · p. 1
- Cell vectors and atomic coordinates do not provide a unique representation, while noisy lattice vectors can trigger discontinuous Niggli reductions and symmetry assignments.Symmetry classification also depends strongly on a tolerance parameter.Source: §I · p. 1
- Earlier crystal fingerprints include element-resolved radial distribution functions and bond characterization matrices, with the former omitting angular information.The bond characterization matrix adds angular information through modified bond-orientational metrics.Source: §I · pp. 1–2
- The proposed crystal fingerprint describes each atom’s environment in real space for an infinite crystal and compares structures through matching atoms with similar environments.This avoids ambiguities of cell-and-coordinate representations.Source: §I · p. 2
II. FINGERPRINT DEFINITION
The method represents each crystal by local atomic fingerprints derived from overlap matrices and defines distance by optimally matching atoms across crystals. The construction is intended to yield a metric, with uniqueness of atomic fingerprints required for the coincidence property.
II. FINGERPRINT DEFINITION
- Each atom’s local fingerprint is formed from eigenvalues of an overlap matrix for a sphere-centered cluster, using Gaussian-type orbitals and a smoothly weighted matrix to avoid boundary discontinuities.The resulting eigenvalues are sorted and zero-padded to keep the fingerprint continuous as atoms enter or leave the sphere.Source: §II · p. 2
- The configurational distance between two crystals is defined by the minimum over atom permutations of the Euclidean differences between their atomic fingerprints.The permutation matches atoms between the two crystals.Source: §II · p. 2
- The distance satisfies the metric properties when atomic fingerprints uniquely distinguish different environments; the paper states uniqueness for fingerprints of length at least three per atom based on its cluster work.Positivity and symmetry follow from the distance definition, while uniqueness supports coincidence and the authors establish the triangle inequality.Source: §II · p. 3
III. CONTRACTED FINGERPRINTS
The ensemble of atomic fingerprints can require substantial storage, particularly for structures with many atoms per unit cell. The paper introduces contraction schemes to reduce the data needed to characterize crystalline structures.
III. CONTRACTED FINGERPRINTS
- An ensemble of atomic fingerprints can require substantial data because each crystal is characterized by fingerprints for all atoms in its unit cell.Typical local fingerprints have at least 50 entries with s orbitals or 200 with s and p orbitals.Source: §III · p. 3
- Storage requirements may become too high for large molecular crystals, motivating schemes that reduce the data needed to characterize a structure.The paper briefly discusses two contraction schemes.Source: §III · p. 3
- The proposed contractions address storage demands while retaining a fingerprint-based characterization of crystalline structures.The stated target includes cases such as large molecular crystals.Source: §III · p. 3
A. Contractions by properties
The property-based contraction groups Gaussian-orbital components according to encoded properties and forms a shorter fingerprint through a generalized eigenvalue problem. The encoded properties can include element and orbital type, with the central atom treated specially in the reported numerical results.
A. Contractions by properties
- The property-based contraction groups Gaussian-orbital components by an encoded property that can represent element identity and s- or p-orbital type.For the reported numerical results, the central atom is treated as special regardless of its chemical type.Source: A. Contractions by properties · p. 3
- A generalized eigenvalue problem using the contracted matrix and its metric tensor produces the contracted fingerprint.The calculation uses the principal vector of the modified overlap matrix as input.Source: A. Contractions by properties · p. 3
- The resulting atomic fingerprint has length (m+1)(lmax+1), shorter than the uncontracted fingerprint.Here m is the number of atomic species and lmax is the maximum angular momentum of the Gaussian orbitals.Source: A. Contractions by properties · p. 3
B. Contractions to form molecular orbitals for molecular crystals
Molecular orbitals provide a more compact basis for molecular-crystal fingerprints, built from local molecular clusters and used to compare each molecule’s environment across structures.
B. Contractions to form molecular orbitals for molecular crystals
- The molecular-orbital basis considerably reduces the data needed to characterize molecular crystals formed from molecules containing many atoms.The method constructs orbitals from a cluster of molecules around each molecule in the unit cell.Source: B. Contractions to form molecular… · p. 3
- Principal vectors belonging to several of the largest overlap-matrix eigenvalues are used for the molecular-orbital contraction.The overlap matrix is formed using Gaussian type orbitals on atoms in the surrounding molecular cluster.Source: B. Contractions to form molecular… · pp. 3–4
- The contracted molecular orbitals are orthogonal, so no metric tensor is required, and their increasing node counts resemble atomic orbitals of increasing angular momentum.The first principal-vector orbital has no nodes; subsequent orbitals have increasing numbers of nodes.Source: B. Contractions to form molecular… · p. 4
- A cutoff based on short-range pseudo-interactions makes molecule-centered overlap matrices continuous under atomic motion and couples molecules sharing a large surface strongly.The interaction sums over atom pairs using their distances and summed van der Waals radii, and vanishes beyond its first zero.Source: B. Contractions to form molecular… · p. 4
- The final overlap-matrix eigenvalues fingerprint each molecule’s environment, and structures are compared across their unit-cell molecules using the Hungarian algorithm.Source: B. Contractions to form molecular… · p. 4
IV. APPLICATION OF FINGERPRINT DISTANCES TO EXPERIMENTAL STRUCTURES
Experimental structures can differ through thermal expansion or cell distortions, so the paper optimizes cell parameters to reduce fingerprint distances while keeping reduced atomic coordinates fixed.
IV. APPLICATION OF FINGERPRINT DISTANCES TO EXPERIMENTAL STRUCTURES
- Thermal expansion, pressure differences, and low-quality x-ray diffraction can produce cell variations in measured structures, making distances between otherwise identical structures nonzero.Source: §IV · p. 4
- The six cell degrees of freedom are optimized with reduced atomic coordinates fixed to minimize distance to a reference configuration.Source: §IV · p. 4
- The fingerprint-distance gradient with respect to lattice vectors is calculated analytically using the Hellmann–Feynman theorem.Source: §IV · p. 4
V. NUMERICAL TESTS
The tests compare fingerprint distances across crystalline and molecular structures, finding that accurate geometry optimization often exposes gaps separating identical from distinct structures. Results also show that performance depends on fingerprint choice, cell size, and structural setting.
V. NUMERICAL TESTS
- For silicon, SFP[s] has length two and does not satisfy the coincidence theorem, whereas SFP[sp] has length at least four and avoids that issue.The authors report that distinct local minima are nevertheless unlikely to lie on the hyperplanes where SFP[s] fingerprints coincide.Source: §V · p. 4
- In silicon, accurate relaxation revealed gaps where sloppy relaxation left them hardly visible; for SiC, our fingerprint showed a gap before relaxation, but all fingerprints did so after accurate relaxation, with BCM weakest.For SiC, the gap disappeared again for large cells, even after high-quality optimization.Source: §V · p. 5
- In Zn2SnO4 and CsPbI3, all fingerprints showed clear gaps, while BCM indicated only a weak gap and Behler showed a pronounced gap for Zn2SnO4 but a blurred one for CsPbI3.Source: §V · p. 5
- For the platinum surface, high-accuracy geometry optimization yielded a large gap with all fingerprints.Source: §V · p. 5
- For formaldehyde and paracetamol, both standard and molecular fingerprints separated identical from distinct structures; the molecular fingerprint used only six principal vectors.The sets comprised 240 formaldehyde structures and 300 paracetamol structures.Source: §V · p. 5
- For 115 ZrO2 database entries, lattice-vector optimization reduced some fingerprint distances from as high as 0.1 to about 1.e-7, but DFT-relaxed energy differences were needed to reveal a clear gap.The structures combined experimental and DFT-optimized data with coordinate and cell noise.Source: §V · p. 5
- For La(Al,Mn)O3 oxygen-vacancy structures, our fingerprint and MINT distinguish the structures, while Oganov and BCM classify them as identical.The test uses symmetrically inequivalent vacancy sites in the La(Al,Mn)O3 supercell.Source: §V · p. 6
VI. CONCLUSIONS
The conclusion presents overlap-matrix atomic fingerprints as a metric-based way to characterize and distinguish crystalline structures. It reports comparable quality to symmetry-function fingerprints at lower calculation cost, and describes extensions to databases, molecular crystals, and machine learning.
VI. CONCLUSIONS
- An ensemble of overlap-matrix atomic fingerprints characterizes crystals and defines configurational distances satisfying metric properties, supporting more reliable distinctions than Oganov and BCM fingerprints.Source: §VI · p. 6
- Scattering fingerprints and symmetry-function fingerprints have similar quality, but both scale cubically with atoms in the cutoff range and the scattering method has a smaller prefactor.The stated cost comparison concerns the matrix diagonalization prefactor versus the three-body terms for symmetry functions.Source: §VI · p. 6
- The method can identify database duplicates through distance gaps and lattice-vector optimization, extend to molecular crystals, and provide local-environment attributes for machine learning.Source: §VI · p. 6
Si30 opt low
The figures show fingerprint correlations across structure-prediction test sets, energy differences, molecular systems, principal vectors, and ZrO2 structures.
Si30 opt low
- Fingerprint correlations are plotted across eight structure-prediction test sets, with short/long fingerprints, orbital choices, and relaxation quality distinguished.“opt high” and “opt low” indicate geometry-relaxation quality.Source: Figure 1 · p. 8
- Energy differences are correlated with fingerprint distances across the eight theoretical test settings shown for Figure 1.Source: Figure 2 · p. 9
- For formaldehyde and paracetamol, the panels relate molecular fingerprint distances to energy differences and to SFP[s] distances.Top panels show energy-difference correlations; bottom panels compare molecular fingerprints with the standard short contracted fingerprint using s orbitals only.Source: Figure 3 · p. 9
- The paracetamol figure colors atoms by the sign of each of the first six principal vectors; the first has no nodes, while higher vectors have increasingly many.Source: Figure 4 · p. 10
- For OQMD ZrO2 structures, Figure 5 plots ordinary against lattice-optimized fingerprint distances and relates ordinary distances to DFT energy differences.Panel (a) colors pairs identified by MINT as identical red and distinct blue; panel (b) shows energy-difference correlation.Source: Figure 5 · p. 11