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Systematic enumeration of crystalline networks

Nature volume 400, pages 644–647 (1999)Cite this article

Abstract

The systematic enumeration of all possible networks of atoms ininorganic structures is of considerable interest. Of particular importance are the 4-connected networks (those in which each atom is connected to exactly four neighbours), which are relevant to a wide range of systems — crystalline elements, hydrates, covalently bonded crystals, silicates and many synthetic compounds. Systematic enumeration is especially desirable in the study of zeolites and related materials, of which there are now 121 recognized structural types1, with several new types being identified every year. But as the number of possible 4-connected three-dimensional networks is infinite, and as there exists no systematic procedure for their derivation, the prediction of new structural types has hitherto relied on empirical methods (see, for example, 2–4). Here we report a partial solution to this problem, basedon recent advances in mathematical tiling theory5,6,7,8. We establish that there are exactly 9, 117 and 926 topological types of, respectively, 4-connected uninodal, binodal and trinodal networks, derived from simple tilings based on tetrahedra. (Here nodality refers to the number of topologically distinct vertices from which the network is composed.) We also show that there are at least 145 more distinct uninodal networks based on a more complex tiling unit. Of the total number of networks that we have derived, only two contain neither three- nor four-membered rings, and most of the binodal and trinodal networks are new.

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Figure 1: Derivation of the Delaney symbol for a tiling.
Figure 2: Vertex figures.
Figure 3: Some of the simple binodal (a, b) and trinodal (c–f) structures enumerated, together with their space groups.

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References

  1. Meier, W. M., Olson, D. H. & Baerlocher, C. Atlas of Zeolite Structure Types (Elsevier, London, 1996); updates available at 〈http://www.iza-sc.ethz.ch/IZA-SC/Atlas/AtlasHome.html〉.

    Google Scholar 

  2. Smith, J. V. Topochemistry of zeolites and related materials. 1. Topology and geometry. Chem. Rev. 88, 149–182 (1988).

    Article  CAS  Google Scholar 

  3. O'Keeffe, M. & Hyde, B. G. Crystal Structures I: Patterns and Symmetry (Monograph, Mineralogical Assoc. of America, Washington DC, 1996).

    MATH  Google Scholar 

  4. Treacy, M. M. J., Randall, K. H., Rao, S., Perry, J. A. & Chadi, D. J. Enumeration of periodic tetrahedral frameworks. Z. Kristallogr. 212, 768–791 (1997).

    CAS  Google Scholar 

  5. Dress, A. W. M., Huson, D. H. & Molnár, E. The classification of face-transitive periodic 3-dimensional tilings. Acta Crystallogr. A 49, 806–817 (1993).

    Article  MathSciNet  Google Scholar 

  6. Dress, A. W. M. in Springer Lecture Notes In Mathematics Vol. 1172, 56–72 (Springer, Göttingen, 1985).

    Google Scholar 

  7. Dress, A. W. M. Presentations of discrete groups, acting on simply connected manifolds, in terms of parametrized systems of Coxeter matrices: a systematic approach. Adv. Math. 63, 196–212 (1987).

    Article  MathSciNet  Google Scholar 

  8. Huson, D. H. Combinatorial tiling theory: theorems-algorithms-visualization.At 〈http://www.mathematik.uni-bielefeld.de/∼delgado/Tiling-Theory〉.

  9. O'Keeffe, M. & Brese, N. E. Uninodal 4-connected 3D nets. I. Nets without 3-rings or 4-rings. Acta Crystallogr. A 48, 663–669 (1992).

    Article  Google Scholar 

  10. Huson, D. H. The generation and classification of tile-k -transitive tilings of the Euclidean plane, the sphere and the hyperbolic plane. Geometriae Dedicata 47, 269–296 (1993).

    Article  MathSciNet  Google Scholar 

  11. Delgado Friedrichs, O., Dress, A. W. M., Müller, A. & Pope, M. T. Polyoxometalates: a class of compounds with remarkable topology. Mol. Eng. 3, 9–28 (1993).

    Article  Google Scholar 

  12. Kepler, J. Harmonices Mundi Libri V Book II, Sections 18–20, pp 51–56 (Linz, 1619).

    Book  Google Scholar 

  13. Heesch, H. Über Raumteilungen. Nachr. Gesellschaft Wiss. Göttingen 1, 35–42 (1934).

    MATH  Google Scholar 

  14. Delgado Friedrichs, O. Euclidicity criteria for three-dimensional branched triangulations.Thesis, Univ. Bielefeld(1994).

  15. Brunner, G. O. & Laves, F. Zum Problem der Koordinazionszahl. Wiss. Z. Tech. Univ. Dresden 20, 387–390 (1971).

    CAS  Google Scholar 

  16. Meier, W. M. & Moeck, H. J. The topology of three-dimensional 4-connected nets: classification of zeolite framework types using coordination sequences. J. Solid State Chem. 27, 349–355 (1979).

    Article  ADS  CAS  Google Scholar 

  17. Wells, A. F. Three-Dimensional Nets and Polyhedra (Wiley, New York, 1977).

    Google Scholar 

  18. Wells, A. F. Further Studies of Three-Dimensional Nets (Am. Crystallographic Assoc. Monogr. No. 8, Polycrystal Book Service, Pittsburgh, 1979).

    Google Scholar 

  19. Grünbaum, B. Uniform tilings of 3-space. Geombinatorics 4, 49–56 (1994).

    MathSciNet  MATH  Google Scholar 

  20. Gramlich-Meier, R. & Meier, W. M. Constituent units and framework conformations in zeolite networks. J. Solid State Chem. 44, 41–49 (1982).

    Article  ADS  CAS  Google Scholar 

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Acknowledgements

We thank M. O'Keeffe for discussions.

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Authors and Affiliations

  1. FSP Mathematisierung-Strukturbildungsprozesse, Universitt Bielefeld, Bielefeld, D-33501, Germany

    Olaf Delgado Friedrichs & Andreas W. M. Dress

  2. Department of Chemical Engineering, City College, CUNY, New York, New York, Convent Avenue at 140th Street, 10031, USA

    Andreas W. M. Dress

  3. Department of Mathematics, Princeton University, Princeton, New Jersey, 08544-1000, USA

    Daniel H. Huson

  4. Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, UK

    Jacek Klinowski

  5. Department of Crystallography, Birkbeck College (University of London), Malet Street, London, WC1E 7HX, UK

    Alan L. Mackay

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  1. Olaf Delgado Friedrichs
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Correspondence to Jacek Klinowski.

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Friedrichs, O., Dress, A., Huson, D. et al. Systematic enumeration of crystalline networks. Nature 400, 644–647 (1999). https://doi.org/10.1038/23210

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