ClusterFinder: a fast tool to find cluster structures from pair distribution function data

Anker, A.S.; Friis-Jensen, U.; Johansen, F.L.; Billinge, S.J.L.; Jensen, K.M.Ø.

doi:10.1107/S2053273324001116

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ISSN: 2053-2733

Volume 80| Part 2| March 2024| Pages 213-220

https://doi.org/10.1107/S2053273324001116

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ClusterFinder: a fast tool to find cluster structures from pair distribution function data

Andy S. Anker,^a Ulrik Friis-Jensen,^a,^b Frederik L. Johansen,^a,^b Simon J. L Billinge ^c ^* and Kirsten M. Ø. Jensen ^a ^*

^aDepartment of Chemistry and Nano-Science Center, University of Copenhagen, 2100 Copenhagen Ø, Denmark, ^bDepartment of Computer Science, University of Copenhagen, 2100 Copenhagen Ø, Denmark, and ^cDepartment of Applied Physics and Applied Mathematics Science, Columbia University, New York, NY 10027, USA
^*Correspondence e-mail: sb2896@columbia.edu, kirsten@chem.ku.dk

Edited by T. E. Gorelik, Helmholtz Centre for Infection Research, Germany (Received 22 November 2023; accepted 1 February 2024; online 29 February 2024)

A novel automated high-throughput screening approach, ClusterFinder, is reported for finding candidate structures for atomic pair distribution function (PDF) structural refinements. Finding starting models for PDF refinements is notoriously difficult when the PDF originates from nanoclusters or small nanoparticles. The reported ClusterFinder algorithm can screen 10⁴ to 10⁵ candidate structures from structural databases such as the Inorganic Crystal Structure Database (ICSD) in minutes, using the crystal structures as templates in which it looks for atomic clusters that result in a PDF similar to the target measured PDF. The algorithm returns a rank-ordered list of clusters for further assessment by the user. The algorithm has performed well for simulated and measured PDFs of metal–oxido clusters such as Keggin clusters. This is therefore a powerful approach to finding structural cluster candidates in a modelling campaign for PDFs of nanoparticles and nanoclusters.

Keywords: pair distribution function analysis; nanoclusters; nanomaterials; screening.

1. Introduction

Throughout the last century, crystallographic methods have played a crucial role in advancing materials science, yet they often struggle when examining nanomaterials with limited long-range order (Billinge & Levin, 2007 ). Total scattering with pair distribution function (PDF) analysis has shown promise for characterizing such nanomaterials (Billinge & Levin, 2007; Christiansen et al., 2020 ). The PDF, derived from the Fourier transform of normalized and corrected X-ray, neutron or electron scattering intensities, offers a real-space representation of interatomic distances in the sample. As the data used in the Fourier transform include both Bragg and diffuse scattering, PDF analysis can be used to characterize the structure of materials with or without long-range atomic order (Egami & Billinge, 2012 ; Christiansen et al., 2020).

The challenge of ab initio structure solution from PDFs has long been pursued (Juhás et al., 2006 , 2008 , 2010 ; Cliffe et al., 2010 ; Cliffe & Goodwin, 2013 ; Anker et al., 2020 ; Kjær et al., 2023 ; Kløve et al., 2023 ). However, success remains limited to rather simple chemical systems like simple inorganic crystals, the C₆₀ buckyball and mono-metallic nanoparticles. In the absence of broadly applicable ab initio structure solution methods, suitable starting models are necessary to refine the PDFs. For crystalline or nanocrystalline materials, such starting models can, in many cases, easily be identified from crystallographic databases. However, this task becomes exceptionally difficult for small clusters and nanomaterials with significant disorder. Recent methods such as ClusterMining (Banerjee et al., 2020 ), StructureMining (Yang et al., 2020 ) and POMFinder (Anker et al., 2024 ) have taken the approach of screening large numbers of structures that are pulled from databases or algorithmically generated. Nonetheless, they are all restrained to the presence of a suitable database of structures or an algorithmic structure generator.

We recently presented a hybrid approach, ML-MotEx (Anker et al., 2022 ), where the user initially selects candidate crystal structures from a crystallographic database. Explainable machine learning is then used to find sub-clusters from the candidate structure that are consistent with the data, which can then be used for further structure refinement and analysis. The approach works well but is slow, taking several minutes for each starting structure. This limits its application to cases where the candidate parent crystal structures are few and obvious to the user. Here, we propose a novel algorithm, ClusterFinder, that follows the same approach of sampling sub-clusters from larger structural candidates, but it uses a non-machine learning direct-scoring approach for identifying high-performing sub-clusters. This speeds up the selection procedure from minutes to seconds, allowing for an automated search for sub-clusters over large numbers of candidate parent structures that can be selected in an automated way from structural databases.

2. Method

The basic strategy for finding clusters from crystalline fragments was described by Anker et al. (2022). We summarize it here. The starting point is an atomic PDF experiment of a sample that contains small clusters, for example a soluble reagent or nanoparticles suspended in a solvent. The atomic arrangement in highly disordered materials can also sometimes be described using cluster structures (Du et al., 2012 ; Castillo-Blas et al., 2020 ; Christiansen et al., 2020). The resulting measured PDF has a small number of peaks confined to the low-r region, indicating the presence of unknown atomic clusters of small size (see Fig. 1).

Figure 1
A simulated PDF for a C₆₀ buckyball from a single unit cell of a C₆₀ crystal structure (Chen & Yamanaka, 2002

). The simulation parameters mimic typical PDF dataset values and can be seen in Section A in the supporting information.

In principle, the data can be fitted using the Debye scattering equation in programs such as DISCUS (Proffen & Neder, 1997 , 1999 ) or DiffPy-CMI (Juhás et al., 2015 ) to understand the structure of the clusters, but this process requires a good initial candidate structure to be given. The main challenge is to find a set of good starting models for the fit. ClusterFinder addresses this need. It reuses the approach taken by ML-MotEx (Anker et al., 2022) where a set of chemically reasonable crystal structures is first identified. From the crystal structures, which are represented using crystallographic information files (CIFs), candidate templates are then cut out. The candidate templates are represented in xyz format (a list of atomic identities and their respective Cartesian coordinates x, y and z). Assuming for now that the cluster present in the experimental data, the target cluster, is contained within the candidate template, the principal goal is to find the subset of occupied atom sites within that template that corresponds to the target cluster. A search over all possible permutations of present versus absent atoms is impossible because of the combinatorics, with 2^N − 1 possibilities for a template of N sites. ML-MotEx uses an explainable machine learning approach to optimize this problem by learning the probabilities that each atom might be present in the target cluster after iterating over a small subset of all the possible permutations. This places the atom sites in a rank-ordered list and makes it easy for the user to select a cut-off for which sites are occupied to generate the target cluster configuration. Of course, the target cluster may not be present in the template and in general there is a further outer loop that needs to be iterated over all possible candidate crystal structures and templates. The ML-MotEx algorithm is too slow to do this over many template candidates and the success of the approach relies on a strong chemical intuition suggesting a small number of candidate structures.

At the heart of the algorithm is the calculation to generate an ordered list of sites based on the probability that they are present in the target cluster. The LIGA algorithm (Juhás et al., 2006, 2008) also scores atoms in a cluster as part of its backtracking cluster reduction step, where poor performing clusters are reduced in size by preferentially removing atoms that are contributing more error to the agreement with the data. The ranking was done using the commonly used PDF weighted profile agreement factor,

$[R_{\rm wp} = \sqrt { {{\sum \nolimits_{i=1}^n [G_{\rm obs} (r_i) - G_{\rm calc} (r_i, P)]^2} \over {\sum \nolimits_{i=1}^n G_{\rm obs} (r_i)^2}}} \times 100 \% , \eqno(1)]$

where G_obs and G_calc are the observed and calculated PDF intensities, respectively, for the set P of model refinement parameters. The sum is over the n points in the PDF.

Taking inspiration from the LIGA algorithm, we attempt an approach of computing the contribution to the fitting error for each atom site in the cluster. We call this the atom-removal error, and denote it for the ith atom by $[\Delta R^i_{\rm wp}]$ . It is computed by evaluating R_wp for the full set of atoms, then recomputing R_wp for the cluster with the ith atom removed and taking the difference. This allows us to identify which atoms contribute the most error to the fit, allowing us to target them for removal. For each computation of R_wp, a scale factor and an isotropic expansion/contraction factor are allowed to be refined to give the best agreement. Atomic displacement parameters (ADPs) were fixed to 0.3 Å² for the metallic atoms and 0.4 Å² for the oxygen atoms. This procedure is extremely rapid and results in a list of atomic sites ranked by $[\Delta R^i_{\rm wp}]$ .

The candidate structure must be large enough to encapsulate the target cluster, but the computational cost scales linearly with the number of atoms of the template structure and so the cluster size chosen can thus be a compromise between time and the cluster structures screened.

To visualize the results, we plot the templates with each atom site colour coded based on its $[\Delta R^i_{\rm wp}]$ . Atom sites with negative (good) $[\Delta R^i_{\rm wp}]$ are coloured yellow and those with positive (bad) $[\Delta R^i_{\rm wp}]$ are coloured blue. The colour coding is further explained in Section B in the supporting information. The approach is illustrated schematically for a trivial example of a small cluster consisting of two atoms in Fig. 2. Note that ClusterFinder only ranks the atoms in the template, and a human input is still needed to determine which atoms to remove in the subsequent task of finding the best cluster candidates. In Fig. 2, it is trivial to remove atoms 3 and 4 from the ClusterFinder output, but this task might not always be trivial and may rely on the chemical intuition of the user. However, it is still extremely valuable because, due to its speed, it can be used to screen large numbers of structures to find the best cluster candidates.

Figure 2
An illustration of the ClusterFinder process. A starting model is provided as input and the R_wp value is calculated by structure refinement. Atoms are iteratively removed from the starting model and the revised model is fitted to the experimental PDF. The atom-removal error $[\Delta R^i_{\rm wp}]$ is calculated by taking the difference between the R_wp values of the full starting model and when the atoms are removed. Atoms are colour coded based on the atom-removal error – yellow indicates a negative $[\Delta R^i_{\rm wp}]$ value (improved fit) while blue signifies a positive $[\Delta R^i_{\rm wp}]$ value (worsened fit).

To test the ClusterFinder approach, we here use it on simulated and experimental PDF data. ClusterFinder provides comparable results to ML-MotEx in quality but orders of magnitude more quickly. The acceleration is sufficient to allow screening of large databases of starting models in minutes. To demonstrate the power of this, we provide five examples where we screen the Inorganic Crystal Structure Database (ICSD, https://icsd.fiz-karlsruhe.de/index.xhtml; Zagorac et al., 2019 ), containing 188 631 structure entries, for a suitable starting model. This is done in a time frame ranging from 3 to 42 min. We expect this to make ClusterFinder highly valuable since, if the target cluster exists anywhere in any known crystal structure, it will automatically be found without any user input at this stage.

3. Results and discussion

3.1. Applying ClusterFinder to extract cluster motifs from simulated PDFs

We first demonstrate ClusterFinder's ability to extract cluster motifs from simulated PDFs. Fig. 3 shows three simulated PDFs, each corresponding to a distinct structure: a decatungstate polyoxometallate cluster from an Na₅(H₇W₁₂O₄₂)(H₂O)₂₀ crystal structure (Redrup & Weller, 2009 ), coloured in blue; a C₆₀ buckyball from the C₆₀ crystal structure (Chen & Yamanaka, 2002 ), coloured in green; and a paratungstate polyoxometallate cluster originated from a (Ba(H₂O)₂{H[N(CH₃)₂]CO}₃)₂(W₁₀O₃₂){H[N(CH₃)₂]CO}₂ crystalline model (Poimanova et al., 2015 ), coloured in red. The values of the simulation parameters used are listed in Section A in the supporting information. Figs. 3(b)–3(d) show the structural templates used by ClusterFinder.

Figure 3
Analysis of simulated PDFs of well known cluster structures. (a) Simulated PDFs of a decatungstate polyoxometallate cluster from the Na₅(H₇W₁₂O₄₂)(H₂O)₂₀ crystal structure (blue) (Redrup & Weller, 2009

), a C₆₀ buckyball from a single unit cell of a C₆₀ crystal structure (green) (Chen & Yamanaka, 2002

) and a paratungstate polyoxometallate cluster obtained from the (Ba(H₂O)₂{H[N(CH₃)₂]CO}₃)₂(W₁₀O₃₂){H[N(CH₃)₂]CO}₂ crystalline model (red) (Poimanova et al., 2015

). Simulation parameters were chosen to mimic typical measured PDF datasets and are listed in Section A in the supporting information. (b)–(d) Results of using ClusterFinder on the three simulated PDFs where the atoms with the (b) 40, (c) 60 and (d) 12 atoms with the lowest $[\Delta R^i_{\rm wp}]$ values have been coloured yellow, while the rest are coloured blue. Section C in the supporting information shows a similar representation but where the atom-removal values are directly shown using a continuous colour bar. Oxygen atoms are coloured red and polyhedra are coloured according to their metal atom centre.

In these tests, the structural templates were constructed using the crystal structures containing each of the cluster structures, and including the minimum number of unit cells needed to include the full cluster (Section C in the supporting information). ClusterFinder outputs a list of atomic sites ranked by the $[\Delta R^i_{\rm wp}]$ value, and we visualize atom sites with negative $[\Delta R^i_{\rm wp}]$ as yellow and those with positive $[\Delta R^i_{\rm wp}]$ as blue. The ranking is here done on the metal atoms, while oxygen atoms are removed if they are beyond a distance threshold of 2.6 Å from any other atom. The resulting visualizations are shown in Figs. 3(b)–3(d), where the atoms with the lowest $[\Delta R^i_{\rm wp}]$ values have been coloured yellow, while the rest are coloured blue. Section C in the supporting information shows a similar representation but where the atom-removal values are directly shown using a continuous colour bar. Oxygen atoms are coloured red and polyhedra are coloured according to their metal atom centre.

ClusterFinder correctly extracted all three cluster structures from their starting model in under a minute using a standard laptop (Intel Core i7-8665U CPU at 1.9/2.11 GHz), demonstrating a significant speed advantage over the ML-MotEx algorithm (Anker et al., 2022), which takes approximately an hour on the same computer. Although ClusterFinder accurately extracts the decatungstate polyoxometallate cluster (blue) and the paratungstate polyoxometallate cluster (red), it does not completely recover the C₆₀ buckyball (green), incorrectly labelling two atoms. The ML-MotEx algorithm also exhibited similar limitations in extracting this structure. Note that while ClusterFinder is faster than ML-MotEx, the latter algorithm is more versatile and has, for example, also been used to determine stacking fault size domain distributions from experimental powder diffraction and PDF data from γ-MnO₂ nanoparticles (Magnard et al., 2022 ).

3.2. Applying ClusterFinder to extract cluster motifs from experimental PDFs

While ClusterFinder's potential to extract cluster motifs from various crystalline supercell structures has been demonstrated with simulated PDFs, it must also work on experimental data. Here we benchmark the performance of ClusterFinder against that of the previously published ML-MotEx algorithm by comparing its performance on the same set of experimental PDFs and clusters.

An experimental PDF was obtained from a solution of 0.05 M ammonium metatungstate hydrate, (NH₄)₆(H₂W₁₂O₄₀)·H₂O in water, which dissolves to form monodisperse α-Keggin clusters (Juelsholt et al., 2019 ). Experimental details can be found in the ML-MotEx paper (Anker et al., 2022). We employed four different crystallographic models to extract templates for ClusterFinder/ML-MotEx as listed in Table 1.

Table 1
Four starting models containing the α-Keggin clusters used with ClusterFinder to extract an α-Keggin cluster

Starting model	Crystal composition	Reference
1	[Hpy]₄H₂[H₂W₁₂O₄₀] (py = pyridine)	Niu et al. (2004)
2	[(CH₃)₄N]₄SiW₁₂O₄₀	Joachim et al. (1981)
3	([(CH₃)₂NH₂]₆{Cu[HCON(CH₃)₂]₄}(GeW₁₂O₄₀)₂)[HCON(CH₃)₂]₂	Niu et al. (2003)
4	[(CH₃)₂NH₂]₃(PW₁₂O₄₀)	Busbongthong & Ozeki (2009)

Again, only a scale factor and an isotropic expansion/contraction factor were refined during the ClusterFinder process. As seen in Fig. 4, both ClusterFinder and ML-MotEx successfully extracted the α-Keggin clusters with few mislabelled atoms for all four starting models. ClusterFinder has slightly more mislabelled atoms than ML-MotEx, but it is orders of magnitude faster, making it an ideal choice for screening larger databases.

Figure 4
Comparisons of the ML-MotEx and ClusterFinder analyses of an experimental PDF obtained from Keggin clusters in solution. Results are given from the ML-MotEx and ClusterFinder methods on a PDF obtained from a solution of ammonium metatungstate hydrate using four different starting models, (1) (Hpy)₄H₂(H₂W₁₂O₄₀) (py = pyridine) (Niu et al., 2004

), (2) [(CH₃)₄N]₄SiW₁₂O₄₀ (Joachim et al., 1981

), (3) ([(CH₃)₂NH₂]₆{Cu[HCON(CH₃)₂]₄}(GeW₁₂O₄₀)₂)[HCON(CH₃)₂]₂ (Niu et al., 2003

) and (4) [(CH₃)₂NH₂]₃(PW₁₂O₄₀) (Busbongthong & Ozeki, 2009

). The 24 [structures (1), (3) and (4)] and 12 [structure (2)] atoms with the lowest atom-removal values have been coloured yellow, while the rest are coloured blue. Oxygen atoms are coloured red.

3.3. Screening the ICSD for a suitable starting model with ClusterFinder

We now use ClusterFinder to scan the whole ICSD for the best-fitting structure models for the experimental PDF obtained from α-Keggin clusters in solution. ClusterFinder uses a single unit cell of each crystal structure (188 631 structures, although we removed unreadable CIFs making it 187 469 structures) in the ICSD as the starting template. To accelerate the ClusterFinder process, only the scale factor was refined, and structures without W, Fe or Mo atoms (158 399 structures), or starting templates with over 1000 atoms (zero structures) were excluded. This left 29 070 candidate structures. For database screening, an isotropic contraction/expansion factor was not refined. Afterwards, the template structures from crystals in the ICSD were ranked according to their average $[\Delta R^i_{\rm wp}]$ value during the ClusterFinder process. The complete computation took 17.5 min (1046 s) on an AMD Ryzen Threadripper 3990X with 64 cores at 2.9/4.3 GHz, or 10 h (34 882 s) on a standard laptop (Intel Core i7-8665U CPU at 1.9/2.11 GHz). Fig. 5 demonstrates that all of the top five crystal structures (Table 2) contained the α-Keggin cluster. This shows ClusterFinder's ability to scan large structural databases effectively, such as the ICSD, for appropriate cluster structures.

Table 2
Crystal composition of the top five candidate crystal structures ranked by ClusterFinder for the PDF obtained from α-Keggin clusters in solution

Ranked structure	Crystal composition	Reference
(I)	[(CH₃)₄N]₆[Cu_0.5(H₂)_0.5O₄W₁₂O₃₆](H₂O)₁₀	Lunk et al. (1993 )
(II)	Cs₅[Cr₃O(OOCH)₆(H₂O)₃](CoW₁₂O₄₀)(H₂O)₂	Uchida et al. (2006 )
(III)	[(CH₃)₄N]₆(H₂W₁₂O₄₀)(H₂O)₉	Asami et al. (1984 )
(IV)	[Al₁₃O₄(OH)₂₄(H₂O)₁₂](H₂W₁₂O₄₀)(OH)(H₂O)_23.12	Son et al. (2003 )
(V)	K₂(H₂O)₄Eu(H₂O)₇[Eu(H₂O)₃HAlW₁₁O₃₉](H₂O)₇	Niu et al. (2013 )

Figure 5
An illustration of how ClusterFinder is used to screen the ICSD for the correct starting model for an experimental PDF obtained from α-Keggin clusters in solution. For each structure in the ICSD, the ClusterFinder procedure is performed, and the atoms are colour coded based on their impact on fit quality using a continuous colour bar. Afterwards, the ICSD structures are sorted according to their average $[\Delta R^i_{\rm wp}]$ values. The five candidate ICSD structures with the lowest average R_wp value are highlighted. The top five candidates are all starting templates exclusively comprising essential cluster structures – clusters in which no atoms need removal and that thereby inherently match their target cluster. An example of an essential α-Keggin structure with additional atoms (non-essential structure) is shown to exemplify that ClusterFinder provides meaningful atomic rankings of non-essential structures. Oxygen atoms are coloured red. Atoms different from W, Fe, Mo or O are omitted for clarity.

ClusterFinder prioritizes starting templates exclusively comprising the essential cluster structure, i.e. clusters in which no atoms need removal and that thereby inherently match their target cluster, over those that contain additional atoms. Consequently, the starting template generation influences the ranking of crystal structures in the ICSD. In instances where exclusively essential clusters are present, the colour coding still reflects the internal atomic ranking, even if all atoms are good and none requires removal. Fig. 5 demonstrates this phenomenon; for instance, starting template (IV) contains only four essential α-Keggin clusters, with no atoms needing removal. However, some atoms are coloured blue, as the colour bar merely signifies the internal atomic ranking. In the case of a starting template containing essential clusters with additional atoms, as seen in Fig. 5, ClusterFinder indicates which atoms require removal.

ClusterFinder can also extract a cluster structure from a crystalline metal oxide structure. The ɛ-Keggin cluster serves as an excellent example of a cluster structure that can be directly cut out from a spinel structure. A PDF of an Al₁₂O₄₀ ɛ-Keggin cluster from the spinel MgAl₂O₄ crystal structure (Ji et al., 2020 ) was simulated with parameters that mimic typical PDF dataset values, as seen in Section A in the supporting information. The PDF and structure are illustrated in Fig. 6. Again, ClusterFinder was used to scan all structures in the ICSD. This time, crystal structures without W, Fe, Mo or Al atoms (143 956 structures) or starting templates with more than 1000 atoms (704 structures) were excluded. After evaluation, 42 809 structures were ranked based on their average $[\Delta R^i_{\rm wp}]$ value found during the ClusterFinder process. The entire procedure takes 42 min (2495 s) on an AMD Ryzen Threadripper 3990X with 64 cores at 2.9/4.3 GHz or 23 h (82 100 s) on a standard laptop (Intel Core i7-8665U CPU at 1.9/2.11 GHz). The top five structures, shown in Fig. 6, are all spinel structures.

Figure 6
An illustration of how ClusterFinder is used to screen the ICSD for the correct starting model for a simulated PDF obtained from an ɛ-Keggin cluster cut out of a spinel crystal structure (coloured light blue in the left of the figure with Mg in the centre). For each structure in the ICSD, the ClusterFinder procedure is performed and the atoms are colour coded based on their impact on the fit quality. Afterwards, the ICSD structures are sorted according to their average $[\Delta R^i_{\rm wp}]$ values during the ClusterFinder process. The five candidates with the lowest R_wp values are highlighted. More extensive views of the PDF fits, including the calculated R_wp values, can be seen in Section D in the supporting information. Atoms different from W, Fe, Mo, Al or O have been omitted for clarity.

We now proceed to apply ClusterFinder to a simulated PDF calculated from the ɛ-Keggin cluster cut out from an ɛ-Keggin crystal structure {here [Al₁₃O₄(OH)₂₄(H₂O)₁₂]₂(V₂W₄O₁₉)₃(OH)₂(H₂O)₂₇; Son & Kwon, 2004 } instead of a cut out from the spinel crystal structure. The ɛ-Keggin obtained in this way is more disordered than that cut out from the spinel crystal structure. The disorder can be seen in both the structures and their PDFs (Figs. 6 and 7), where the PDF simulated from the spinel-derived ɛ-Keggin (Fig. 6) exhibits sharper peaks than the PDF simulated from the ɛ-Keggin cluster cut out of the [Al₁₃O₄(OH)₂₄(H₂O)₁₂]₂(V₂W₄O₁₉)₃(OH)₂(H₂O)₂₇ crystal structure (Son & Kwon, 2004) (Fig. 7). Again, we use ClusterFinder on all ICSD structures containing W, Fe, Mo or Al atoms one by one. Afterwards, it ranks the structures based on their average $[\Delta R^i_{\rm wp}]$ value obtained during the ClusterFinder process. Fig. 7 and Table 4 show that the top five structures mainly contain ɛ-Keggin clusters or are variants of the spinel structure [structures (III) and (V)]. While α-Keggin and ɛ-Keggin clusters are very similar and only distinct in the different rotational orientations of their four M₃O₁₃ units, ClusterFinder is able to differentiate between them in starting template structures (I) and (II) where the α-Keggin motif is removed (blue) and the ɛ-Keggin motifs are kept (yellow).

Figure 7
An illustration of how ClusterFinder is used to screen the ICSD for the correct starting model for a simulated PDF obtained from an ɛ-Keggin cluster cut out of an ɛ-Keggin crystal structure (coloured light blue in the left of the figure). For each structure in the ICSD, the ClusterFinder procedure is performed and the atoms are colour coded based on their impact on the fit quality. Afterwards, the ICSD structures are sorted according to their average $[\Delta R^i_{\rm wp}]$ values during the ClusterFinder process. The five candidates with the lowest R_wp value are highlighted. More extensive views of the PDF fits, including the calculated R_wp values, can be seen in Section E in the supporting information. Oxygen atoms are coloured red. Other atoms than W, Fe, Mo, Al or O have been omitted for clarity.

ClusterFinder can, moreover, discern between the more ordered spinel-obtained motifs (Fig. 6 and Table 3) and the more distorted Keggin crystal structure (Fig. 7 and Table 4), which demonstrates that it is sensitive to minor changes in the PDF. This highlights the level of detailed description attained in this modelling approach.

Table 3
Crystal composition of the top five candidate crystal structures ranked by ClusterFinder for the simulated PDF from the Al₁₂O₄₀ ɛ-Keggin cluster cut out from the spinel MgAl₂O₄ crystal structure

Ranked structure	Crystal composition	Reference
(I)	NiAl₂O₄	Vegard & Borlaug (1943 )
(II)	MgAl₂O₄	Zorina & Kvitka (1968 )
(III)	ZnAl₂O₄	Holgersson (1927 )
(IV)	ZnAl₂O₄	Vegard & Borlaug (1943)
(V)	ZnAl₂O₄	Saalfeld (1964 )

Table 4
Crystal composition of the top five candidate crystal structures calculated by ClusterFinder for the simulated PDF from the ɛ-Keggin cluster cut out of the Al₁₂O₄₀ [Al₁₃O₄(OH)₂₄(H₂O)₁₂]₂(V₂W₄O₁₉)₃(OH)₂(H₂O)₂₇ crystal structure (Son & Kwon, 2004)

Ranked structure	Crystal composition	Reference
(I)	[Al₁₃O₄(OH)₂₄(H₂O)₁₂](H₂W₁₂O₄₀)(OH)(H₂O)_23.12	Son et al. (2003)
(II)	[Al₁₃O₄(OH)₂₄(H₂O)₁₂](CoW₁₂O₄₀)(OH)(H₂O)₂₀	Son et al. (2003)
(III)	Ca₂Mg₂Fe₂[Al₁₄O₃₁(OH)](Al₂O)(Al)[Al(OH)]	Rastsvetaeva et al. (2010 )
(IV)	[(GeO₄)Al₁₂(OH)₂₄(H₂O)₁₂](SeO₄)₄(H₂O)₁₄	Lee et al. (2001 )
(V)	(Al₂O₃)₁₃(SO₃)₆(H₂O)₇₉	Nordstrom (1982 )

In Sections F and G in the supporting information, we present two similar examples in which we rank the ICSD structures according to experimental datasets obtained from ionic [Bi₃₈O₄₅] clusters and ceria (CeO₂) nanoparticles. We find that the highest ranked structures from the [Bi₃₈O₄₅] cluster example are δ-Bi₂O₃ crystal structures, as previously observed by Weber et al. (2017 ). For the ceria nanoparticles, the highest ranked structures correspond to bixbyite-type structures, which are related to the fluorite-type structure that CeO₂ would be expected to take. This demonstrates that, while ClusterFinder often provides results closely related to the true chemical solution, validation and considerations of structure relations are still required in the data analysis process.

4. Conclusions

We have introduced a new automated structure selection approach called ClusterFinder for identifying suitable starting models for analysis and refinement of PDFs from nanoclusters. The premise of ClusterFinder is that the structure of a nanocluster can probably be described as a fragment of an already published crystal structure, and it thus screens crystal structures and identifies fragments for further analysis. The structure found by ClusterFinder is not necessarily a unique solution to the PDF, but ClusterFinder's automated process ensures a systematic and extensive screening of a range of possible structures.

ClusterFinder is inspired by our previously developed algorithms, LIGA and ML-MotEx, but is significantly faster, facilitating screening of large databases for cluster identification in minutes. Our study demonstrates ClusterFinder's efficacy as a robust tool for extracting appropriate starting models from extensive structural databases like the ICSD. By applying ClusterFinder to PDFs from various nanoclusters, such as α-Keggin clusters, ɛ-Keggin clusters, ionic [Bi₃₈O₄₅] clusters and ceria nanoparticles, we have showcased its abilities in effectively ranking and selecting the most relevant structure models based on fit quality.

All the data supporting this study are available either within the paper, as supporting information or on the associated GitHub to the paper, https://github.com/AndySAnker/ClusterFinder. The code supporting this study is also available on the same associated GitHub.

5. Related literature

For further literature related to the supporting information, see Anker et al. (2021 ), Artini et al. (2014 ), Chakraborty et al. (2006 ), Coduri et al. (2013 ), Estes et al. (2016 ), Juhás et al. (2013 ), Labidi et al. (2008 ), Rademacher et al. (2001 ), Radosavljević-Evans et al. (2002 ), Sasaki et al. (2004 ) and Yang et al. (2014 ).

Supporting information

Additional background. DOI: https://doi.org/10.1107/S2053273324001116/tw5008sup1.pdf

Acknowledgements

We acknowledge the MAX IV Laboratory for time on beamline DanMAX under Proposal 20200731. We acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association HGF, for the provision of experimental facilities. Parts of this research were carried out on beamline P02.1 at PETRA III, and we thank Martin Etter and Jozef Bednarcik for assistance in using the beamline. Author contributions are as follows: ASA contributed to all aspects of the paper; ASA, UFJ and FLJ wrote the code; KMØJ and SJLB procured funding; SJLB and KMØJ supervised the project; all authors contributed to the writing of the manuscript. The authors declare no competing interests.

Funding information

This work is part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme (grant agreement No. 804066). Work in the Billinge group was supported by the US National Science Foundation (grant No. DMREF-1922234). We are grateful to the Villum Foundation for financial support through a Villum Young Investigator grant (No. VKR00015416). Funding from the Danish Ministry of Higher Education and Science through the SMART Lighthouse is gratefully acknowledged.

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research papers\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

ClusterFinder: a fast tool to find cluster structures from pair distribution function data

1. Introduction

2. Method

3. Results and discussion

3.1. Applying ClusterFinder to extract cluster motifs from simulated PDFs

3.2. Applying ClusterFinder to extract cluster motifs from experimental PDFs

3.3. Screening the ICSD for a suitable starting model with ClusterFinder

4. Conclusions

5. Related literature

Supporting information

Acknowledgements

Funding information

References

research papers