Proposed definition of crystal substructure and substructural similarity

Lusann Yang, Stephen Dacek, and Gerbrand Ceder

Phys. Rev. B 90, 054102 – Published 5 August 2014

Abstract

There is a clear need for a practical and mathematically rigorous description of local structure in inorganic compounds so that structures and chemistries can be easily compared across large data sets. Here a method for decomposing crystal structures into substructures is given, and a similarity function between those substructures is defined. The similarity function is based on both geometric and chemical similarity. This construction allows for large-scale data mining of substructural properties, and the analysis of substructures and void spaces within crystal structures. The method is validated via the prediction of Li-ion intercalation sites for the oxides. Tested on databases of known Li-ion-containing oxides, the method reproduces all Li-ion sites in an oxide with a maximum of 4 incorrect guesses 80% of the time.

Received 10 May 2014
Revised 22 July 2014

DOI:https://doi.org/10.1103/PhysRevB.90.054102

Authors & Affiliations

Lusann Yang, Stephen Dacek, and Gerbrand Ceder^*

Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

^*gceder@mit.edu

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Vol. 90, Iss. 5 — 1 August 2014

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Images

Figure 1
Variations on the perovskite crystal structure. Three perovskite crystal structures are shown. The first structure, ${SrTiO}_{3}$ , is the perovskite crystal structure. The next two structures are variations on perovskite. The second features a slight orthorhombic distortion and alternating Fe and Mo ions, while the third features distorted octahedra. While the octahedral environments vary in both chemistry and geometry in all three structures, they are nonetheless similar.
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Figure 2
The Voronoi polyhedron around a central ion is shown in gray. Faces of the Voronoi polyhedron are generated by the perpendicular bisectors of the lines between the central ion and all other ions in the crystal structure. Each face of the Voronoi polyhedron is thus caused by the existence of a neighboring ion $y$ . The neighboring ion whose polyhedron face subtends the largest solid angle $Ω_{\max}$ has weight 1; all other neighboring ions have weights proportional to the angle subtended by their polyhedron faces.
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Figure 3
The Li site identification rate. The solid line shows the number of incorrect guesses before identifying all the Li sites in a structure via the substructural similarity method; the dotted line shows the number of incorrect guesses using the radius of the site. Sub-structural similarity finds all the Li sites within a crystal structure with 9 incorrect guesses 90% of the time. The striped region on the right represents the 5% of Li sites that are not within $0.5 Å$ of a Voronoi point, and thus cannot be found by this algorithm.
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Figure 4
The receiver operating characteristic for Li site classification. The $x$ axis depicts the false positive fraction, or fraction of non-Li-site Voronoi points that were mistakenly identified as Li sites. The $y$ axis depicts the true positive fraction, or the fraction of Li sites that were correctly identified as Li sites. The black line represents classification by substructural similarity; the dotted line represents classification by site radius.
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Figure 5
Performance of Li site prediction by compound complexity. The number of incorrect guesses required to find all the Li sites is plotted versus the complexity of the oxide host. The complexity of the host is given by the number of symmetrically distinct sites in the host structure. Ranking by substructural similarity performs as well on simple oxide structures as it does on more complex oxides.
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Figure 6
Distance distribution of Li sites and number of guesses. The $x$ axis depicts the distance as calculated by the similarity metric between an Li site in the test set and the closest Li site in the training set. The blue $y$ axis on the left-hand side depicts the distribution of Li sites versus distance; the red $y$ axis on the right-hand side depicts the average number of guesses at a given distance. An ideal ranking system would increase the lag between the blue and the red lines, finding 100% of the Li sites before requiring more than 1 guess.
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