We have analyzed structural motifs in the Deem database of hypothetical zeolites to investigate whether the structural diversity found in this database can be well-represented by classical descriptors, such as distances, angles, and ring sizes, or whether a more general representation of the atomic structure, furnished by the smooth overlap of atomic position (SOAP) method, is required to capture accurately structure–property relations. We assessed the quality of each descriptor by machine-learning the molar energy and volume for each hypothetical framework in the dataset. We have found that a SOAP representation with a cutoff length of 6 Å, which goes beyond near-neighbor tetrahedra, best describes the structural diversity in the Deem database by capturing relevant interatomic correlations. Kernel principal component analysis shows that SOAP maintains its superior performance even when reducing its dimensionality to those of the classical descriptors and that the first three kernel principal components capture the main variability in the dataset, allowing a 3D point cloud visualization of local environments in the Deem database. This “cloud atlas” of local environments was found to show good correlations with the contribution of a given motif to the density and stability of its parent framework. Local volume and energy maps constructed from the SOAP/machine learning analyses provide new images of zeolites that reveal smooth variations of local volumes and energies across a given framework and correlations between the contributions to volume and energy associated with each atom-centered environment.
REFERENCES
1.
S. M.
Auerbach
, K. A.
Carrado
, and P. K.
Dutta
, Handbook of Zeolite Science and Technology
(CRC Press
, 2003
).2.
Ch.
Baerlocher
, L. B.
McCusker
, and D. H.
Olson
, Atlas of Zeolite Framework Types
(Elsevier
, 2007
).3.
M. W.
Deem
, R.
Pophale
, P. A.
Cheeseman
, and D. J.
Earl
, “Computational discovery of new zeolite-like materials
,” J. Phys. Chem. C
113
, 21353
–21360
(2009
).4.
R.
Pophale
, P. A.
Cheeseman
, and M. W.
Deem
, “A database of new zeolite-like materials
,” Phys. Chem. Chem. Phys.
13
, 12407
–12412
(2011
), the SLC-PCOD database can be accessed at http://www.hypotheticalzeolites.net/DATABASE/DEEM/DEEM_PCOD/index.php.5.
M. M. J.
Treacy
, K. H.
Randall
, S.
Rao
, J. A.
Perry
, and D. J.
Chadi
, “Enumeration of periodic tetrahedral frameworks
,” Z. Kristallogr. - Cryst. Mater.
212
, 768
–791
(1997
).6.
M. M. J.
Treacy
, I.
Rivin
, E.
Balkovsky
, K. H.
Randall
, and M. D.
Foster
, “Enumeration of periodic tetrahedral frameworks. II. Polynodal graphs
,” Microporous Mesoporous Mater.
74
, 121
–132
(2004
).7.
Y.
Li
, J. H.
Yu
, and R. R.
Xu
, “Criteria for zeolite frameworks realizable for target synthesis
,” Angew. Chem., Int. Ed.
52
, 1673
–1677
(2013
).8.
L. C.
Lin
, A. H.
Berger
, R. L.
Martin
, J.
Kim
, J. A.
Swisher
, K.
Jariwala
, C. H.
Rycroft
, A. S.
Bhown
, M. W.
Deem
, M.
Haranczyk
, and B.
Smit
, “In silico screening of carbon-capture materials
,” Nat. Mater.
11
, 633
–641
(2012
).9.
A.
Lupulescu
and J. D.
Rimer
, “In situ imaging of silicalite-1 surface growth reveals the mechanism of crystallization
,” Science
344
, 729
–732
(2014
).10.
M.
Kumar
, M. K.
Choudhary
, and J. D.
Rimer
, “Transient modes of zeolite surface growth from 3D gel-like islands to 2D single layers
,” Nat. Commun.
9
, 2129
(2018
).11.
X.
Zhu
, M. G.
Goesten
, A. J. J.
Koekkoek
, B.
Mezari
, N.
Kosinov
, G.
Filonenko
, H.
Friedrich
, R.
Rohling
, B. M.
Szyja
, J.
Gascon
, F.
Kapteijn
, and E. J. M.
Hensen
, “Establishing hierarchy: The chain of events leading to the formation of silicalite-1 nanosheets
,” Chem. Sci.
7
, 6506
–6513
(2016
).12.
V. A.
Blatov
, G. D.
Ilyushin
, and D. M.
Proserpio
, “The zeolite conundrum: Why are there so many hypothetial zeolites and so few observed? A possible answer from the zeolite-type frameworks perceived as packings of tiles
,” Chem. Mater.
25
, 412
–424
(2013
).13.
A. P.
Bartók
, R.
Kondor
, and G.
Csányi
, “On representing chemical environments
,” Phys. Rev. B
87
, 184115
(2013
).14.
S.
De
, A. P.
Bartók
, G.
Csányi
, and M.
Ceriotti
, “Comparing molecules and solids across structural and alchemical space
,” Phys. Chem. Chem. Phys.
18
, 13754
–13769
(2016
).15.
N. J.
Henson
, A. K.
Cheetham
, and J. D.
Gale
, “Theoretical calculations on silica frameworks and their correlation with experiment
,” Chem. Mater.
6
, 1647
–1650
(1994
).16.
I.
Petrovic
, A.
Navrotsky
, M. E.
Davis
, and S. I.
Zones
, “Thermochemical study of the stability of frameworks in high silica zeolites
,” Chem. Mater.
5
, 1805
–1813
(1993
).17.
Database of Zeolite Structures – Explanatory Notes – Composite Building Units,
International Zeolite Association
, 2018
.18.
M. J.
Sanders
, M.
Leslie
, and C. R. A.
Catlow
, “Interatomic potentials for SiO2
,” J. Chem. Soc., Chem. Commun.
1984
, 1271
–1273
.19.
K.-P.
Schröder
, J.
Sauer
, M.
Leslie
, C.
Richard
, A.
Catlow
, and J. M.
Thomas
, “Bridging hydroxyl groups in zeolitic catalysts: A computer simulation of their structure, vibrational properties and acidity in protonated faujasites (H-Y zeolites)
,” Chem. Phys. Lett.
188
, 320
–325
(1992
).20.
J. D.
Gale
and A. L.
Rohl
, “The general utility lattice program (GULP)
,” Mol. Simul.
29
, 291
–341
(2003
).21.
S. I.
Zones
, “Translating new materials discoveries in zeolite research to commercial manufacture
,” Microporous Mesoporous Mater.
144
, 1
–8
(2011
).22.
D. J.
Earl
and M. W.
Deem
, “Toward a database of hypothetical zeolite structures
,” Ind. Eng. Chem. Res.
45
, 5449
–5454
(2006
).23.
R. A.
Curtis
and M. W.
Deem
, “A statistical mechanics study of ring size, ring shape, and the relation to pores found in zeolites
,” J. Phys. Chem. B
107
, 8612
–8620
(2003
).24.
M.
Ceriotti
, “Unsupervised machine learning in atomistic simulations, between predictions and understanding
,” J. Chem. Phys.
150
, 150901
(2019
).25.
A. P.
Bartók
, S.
De
, C.
Poelking
, N.
Bernstein
, J. R.
Kermode
, G.
Csányi
, and M.
Ceriotti
, “Machine learning unifies the modeling of materials and molecules
,” Sci. Adv.
3
, e1701816
(2017
).26.
L.
Ward
, R.
Liu
, A.
Krishna
, V. I.
Hegde
, A.
Agrawal
, A.
Choudhary
, and C.
Wolverton
, “Including crystal structure attributes in machine learning models of formation energies via Voronoi tessellations
,” Phys. Rev. B
96
, 024104
(2017
).27.
F.
Faber
, A.
Lindmaa
, O.
Anatole von Lilienfeld
, and R.
Armiento
, “Crystal structure representations for machine learning models of formation energies
,” Int. J. Quantum Chem.
115
, 1094
–1101
(2015
).28.
F. A.
Faber
, A.
Lindmaa
, O.
Anatole von Lilienfeld
, and R.
Armiento
, “Machine learning energies of 2 million elapsolite ABC2D6 crystals
,” Phys. Rev. Lett.
117
, 135502
(2016
).29.
F. A.
Faber
, L.
Hutchison
, B.
Huang
, J.
Gilmer
, S. S.
Schoenholz
, G. E.
Dahl
, O.
Vinyals
, S.
Kearnes
, P. F.
Riley
, and O.
Anatole von Lilienfeld
, “Prediction errors of molecular machine learning models lower than hybrid DFT error
,” J. Chem. Theory Comput.
13
, 5255
–5264
(2017
).30.
K.
Hansen
, G.
Montavon
, F.
Biegler
, S.
Fazli
, M.
Rupp
, M.
Scheffler
, O.
Anatole von Lilienfeld
, A.
Tkatchenko
, and K.-R.
Müller
, “Assessment and validation of machine learning models for predicting moleclar atomization energies
,” J. Chem. Theory Comput.
9
, 3404
–3419
(2013
).31.
K.
Hansen
, F.
Biegler
, R.
Ramakrishnan
, W.
Pronobis
, O.
Anatole von Lilienfeld
, K.-R.
Müller
, and A.
Tkatchenko
, “Machine learning predictions of molecular properties: Accurate many-body potentials and nonlocality in chemical space
,” J. Phys. Chem. Lett.
6
, 2326
–2331
(2015
).32.
K. T.
Schütt
, H.
Glawe
, F.
Brockherde
, A.
Sanna
, K. R.
Müller
, and E. K. U.
Gross
, “How to represent crystal structures for machine learning: Towards fast prediction of electronic properties
,” Phys. Rev. B
89
, 205118
(2014
).33.
M.
Rupp
, A.
Tkatchenko
, K.-R.
Müller
, and O.
Anatole von Lilienfeld
, “Fast and accurate modeling of molecular atomization energies with machine learning
,” Phys. Rev. Lett.
108
, 058301
(2012
).34.
O.
Anatole von Lilienfeld
, R.
Ramakrishnan
, M.
Rupp
, and A.
Knoll
, “Fourier series of atomic radial distribution functions: A molecular fingerprint for machine learning models of quantum chemical properties
,” Int. J. Quantum Chem.
115
, 1084
–1093
(2015
).35.
R.
Ramprasad
, R.
Batra
, G.
Pilania
, A.
Mannodi-Kanakkithodi
, and C.
Kim
, “Machine learning in materials informatics: Recent applications and prospects
,” npj Comput. Mater.
3
, 54
(2017
).36.
O.
Isayev
, C.
Oses
, C.
Toher
, E.
Gossett
, S.
Curtarolo
, and A.
Tropsha
, “Universal fragment descriptors for predicting properties of inorganic crystals
,” Nat. Commun.
8
, 15679
(2017
).37.
M.
Gastegger
, L.
Schwiedrzik
, M.
Bittermann
, F.
Berzsenyi
, and P.
Marquetand
, “wACSF—weighted atom-centered symmetry functions as descriptors in machine learning potentials
,” J. Chem. Phys.
148
, 241709
(2018
).38.
J.
Behler
, “Atom-centered symmetry functions for constructing high-dimensional neural network potentials
,” J. Chem. Phys.
134
, 074106
(2011
).39.
A. P.
Bartók
, M. C.
Payne
, R.
Kondor
, and G.
Csányi
, “Gaussian approximation potentials: The accuracy of quantum mechanics, without the electrons
,” Phys. Rev. Lett.
104
, 136403
(2010
).40.
A.
Glielmo
, C.
Zeni
, and A.
De Vita
, “Efficient nonparametric n-body force fields from machine learning
,” Phys. Rev. B
97
, 184307
(2018
).41.
M. J.
Willatt
, F.
Musil
, and M.
Ceriotti
, “Atom-density representations for machine learning
,” J. Chem. Phys.
150
, 154110
(2019
).42.
R.
Drautz
, “Atomic cluster expansion for accurate and transferable interatomic potentials
,” Phys. Rev. B
99
, 014104
(2019
).43.
S.
Yang
, M.
Lach-hab
, I. I.
Vaisman
, and E.
Blaisten-Barojas
, “Identifying zeolite frameworks with a machine learning approach
,” J. Phys. Chem. C
113
, 21721
–21725
(2009
).44.
J. D.
Evans
and F.-X.
Coudert
, “Predicting the mechanical properties of zeolite frameworks by machine learning
,” Chem. Mater.
29
, 7833
–7839
(2017
).45.
A. P.
Bartók
and G.
Csányi
, “Gaussian approximation potentials: A brief tutorial introduction
,” Int. J. Quantum Chem.
115
, 1051
–1057
(2015
).46.
A. P.
Bartók
, W. J.
Szlachta
, and G.
Csányi
, “Accuracy and transferability of Gaussian approximation potential models for tungsten
,” Phys. Rev. B
90
, 104108
(2014
).47.
A. P.
Bartók
, J.
Kermode
, N.
Bernstein
, and G.
Csányi
, “Machine learning a general-purpose interatomic potential for silicon
,” Phys. Rev. X
8
, 041048
(2018
).48.
J.-B.
Maillet
, C.
Denoual
, and G.
Csányi
, “Machine-learning based potential for iron: Plasticity and phase transition
,” AIP Conf. Proc.
1979
, 050011
(2018
).49.
V. L.
Deringer
and G.
Csányi
, “Machine learning based interatomic potential for amorphous carbon
,” Phys. Rev. B
95
, 094203
(2017
).50.
D.
Dragoni
, T. D.
Daff
, G.
Csányi
, and N.
Marzari
, “Achieving DFT accuracy with a machine-learning interatomic potential: Thermomechanics and defects in bcc ferromagnetic iron
,” Phys. Rev. Mater.
2
, 013808
(2018
).51.
F.
Musil
, S.
De
, J.
Yang
, J. E.
Campbell
, G. M.
Day
, and M.
Ceriotti
, “Machine learning for the structure–energy–property landscapes of molecular crystals
,” Chem. Sci.
9
, 1289
–1300
(2018
).52.
B. A.
Helfrecht
, P.
Gasparotto
, F.
Giberti
, and M.
Ceriotti
, “Atomic motif recognition in (bio)polymers: Benchmarks from the protein data bank
,” Front. Mol. Biosci.
6
, 24
(2019
).53.
J. V.
Smith
, “Structural classification of zeolites
,” Mineral. Soc. Am.
1
, 281
(1963
).54.
S.
Le Roux
and P.
Jund
, “Ring statistics analysis of topological networks: New approach and application to amorphous GeS2 and SiO2 systems
,” Comput. Mater. Sci.
49
, 70
(2010
).55.
A. M.
Stoneham
and J. H.
Harding
, “Interatomic potentials in solid state chemistry
,” Annu. Rev. Phys. Chem.
37
, 53
–80
(1986
).56.
H.
Balamane
, T.
Halicioglu
, and W. A.
Tiller
, “Comparative study of silicon empirical interatomic potentials
,” Phys. Rev. B
46
, 2250
–2279
(1992
).57.
S. V.
King
, “Rings configurations in a random network model of vitreous silica
,” Nature
213
, 1112
(1967
).58.
L.
Guttman
, “Rings structure of the crystalline and amorphous forms of silicon dioxide
,” J. Non-Cryst. Solids
116
, 145
(1990
).59.
C. S.
Marians
and L. W.
Hobbs
, “Network properties of crystalline polymorphs of silica
,” J. Non-Cryst. Solids
124
, 242
–253
(1990
).60.
D. S.
Franzblau
, “Computation of ring statistics for network models of solids
,” Phys. Rev. B
44
, 4925
–4930
(1991
).61.
See http://rings-code.sourceforge.net/ for R.I.N.G.S. code.
62.
B.
Huang
and O.
Anatole von Lilienfeld
, “Communication: Understanding molecular representations in machine learning: The role of uniqueness and target similarity
,” J. Chem. Phys.
145
, 161102
(2016
).63.
A. J.
Smola
and B.
Schökopf
, “Sparse greedy matrix approximation for machine learning
,” in Proceedings of the Seventeenth International Conference on Machine Learning, ICML ’00
(Morgan Kaufmann Publishers, Inc.
, San Francisco, CA, USA
, 2000
), pp. 911
–918
.64.
65.
M.
Ceriotti
, M. J.
Willatt
, and G.
Csányi
, “Machine learning of atomic-scale properties based on physical principles
,” in Handbook of Materials Modeling
, edited by W.
Andreoni
and S.
Yip
(Springer
, Cham
, 2018
).66.
Y.
Eldar
, M.
Lindenbaum
, M.
Porat
, and Y. Y.
Zeevi
, “The farthest point strategy for progressive image sampling
,” IEEE Trans. Image Process.
6
, 1305
–1315
(1997
).67.
M.
Ceriotti
, G. A.
Tribello
, and M.
Parrinello
, “Demonstrating the transferability and the descriptive power of sketch-map
,” J. Chem. Theory Comput.
9
, 1521
–1532
(2013
).68.
G.
Imbalzano
, A.
Anelli
, D.
Giofré
, S.
Klees
, J.
Behler
, and M.
Ceriotti
, “Automatic selection of atomic fingerprints and reference configurations for machine-learning potentials
,” J. Chem. Phys.
148
, 241730
(2018
).69.
For some of the ring descriptors, the dataset contains fewer than 2000 unique feature vectors, in which case only the unique representations were considered as representative environments.
70.
B.
Schölkopf
, A. J.
Smola
, and K.-R.
Müller
, “Nonlinear component analysis as a kernel eigenvalue problem
,” Neural Compututation
10
, 1299
–1319
(1998
).71.
D. M.
Wilkins
, A.
Grisafi
, Y.
Yang
, K. U.
Lao
, R. A.
DiStasio
, and M.
Ceriotti
, “Accurate molecular polarizabilities with coupled cluster theory and machine learning
,” Proc. Natl. Acad. Sci. U. S. A.
116
, 3401
–3406
(2019
).72.
M. J.
Willatt
, F.
Musil
, and M.
Ceriotti
, “Feature optimization for atomistic machine learning yields a data-driven construction of the periodic table of the elements
,” Phys. Chem. Chem. Phys.
20
, 29661
–29668
(2018
).73.
S. M.
Auerbach
, W.
Fan
, and P. A.
Monson
, “Modeling the assembly of nanoporous silica materials
,” Int. Rev. Phys. Chem.
34
, 35
–70
(2015
).74.
C.
Bores
, S. M.
Auerbach
, and P. A.
Monson
, “Modeling the role of excluded volume in zeolite structure direction
,” J. Phys. Chem. Lett.
9
, 3703
–3707
(2018
).© 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.