The systematic enumeration of all possible networks of atoms in inorganic structures is a matter of considerable interest. For example, 4connected networks (i.e., networks in which each atom is connected to exactly four neighbours) occur in crystalline elements, hydrates, covalently bonded crystals, silicates and many synthetic compounds. Unfortunately, the problem is fraught with difficulties, and since the number of 4connected 3D networks is infinite and there is no systematic procedure for their derivation, the results reported so far have been obtained by empirical methods.
In collaboration with
Dr. Olaf Delgado Friedrichs and
Professor Daniel H. Huson (both at
University of Bielefeld, now at the
University of Tübingen), Professor Andreas W. M. Dress
(University of Bielefeld), and Professor Alan L. Mackay, we have reported a completely new approach to the problem of systematic enumeration based on advances in mathematical tiling
theory.[1] We have shown that there are exactly 9, 117 and 926 topological types of 4connected uninodal, binodal and trinodal networks, respectively, derived from
simple tilings (tilings with vertex figures which are tetrahedra), and at least 145 additional uninodal networks derived from
quasisimple tilings (the vertex figures of which are derived from tetrahedra, but contain double edges). Most of the binodal and trinodal networks are new. Using this method it is possible, in principle, to enumerate every possible periodic threedimensional network, subject only to the availability of computer resources. Our results and methods are applicable to the structure of zeolites, silicate networks substituted with heteroatoms (such as Al, Ga, Be etc.), aluminophosphates and related materials, nitrides, chalcogenides and halides, to 3 and 4connected carbon networks and to structures such as ice. Finally, bubbles found in foams are polyhedra in simple tilings. This means that every periodic foam with 1, 2 and 3 kinds of vertex will be found among our total of 1052 (= 9 + 117 + 926) simple
tilings.
Given the enormously wide industrial applications of zeolites as molecular sieves, ion exchangers, catalysts and catalyst supports, as well as their intrinsic academic interest, a great amount of work has been done on the characterization of known zeolitic structures, of which 121 distinct structural types have been
identified.[2] The design of new zeolite frameworks is also a matter of considerable practical importance for two reasons. Firstly, a list of feasible hypothetical structures would enable the design strategies leading to the synthesis of at least some of them. Secondly, Xray and neutron diffraction patterns generated for such hypothetical structures will be of considerable help in determining the structures of new zeolitic materials.
Enumeration of hypothetical zeolitic
structures[3] is closely related to the work of
Wells[46] on threedimensional nets and polyhedra. Smith and
collaborators,[724] Alberti,[25]
Sato,[2629] Sherman and
Bennett,[30] Barrer and
Villiger,[31] O’Keeffe and
collaborators[3238] and Akporiaye and
Price[39] found many possible new structures by linking together structural subunits in new ways, while more recent work involved computer search
algorithms.[38,40,41] Thus Treacy et al.[40] described 150 coordination sequences of uninodal structures generated by simulated annealing. However, as the number of possible threedimensional frameworks made from tetrahedral building blocks is infinite, such approach can by its very nature generate only a small fraction of the possible networks. By contrast, subject to restraints on unit cell size and the number of inequivalent sites, tiling theory can generate
all possible zeolitic structures.
Not all systematically enumerated zeolitic structures are
chemically feasible. In collaboration with Dr. Robert G. Bell and
Martin D. Foster at the
Davy Faraday Research
Laboratory, The Royal Institution of Great
Britain, we have used molecular modelling techniques to calculate the structure, lattice energy, framework density and other structural parameters of the hypothetical frameworks enumerated via the tiling theory. An automated procedure uses these quantities, and correlations between them, to make a selection of feasible zeolitic structures. Each framework was assumed to have the chemical formula SiO2 and was optimized using a standard lattice energy minimization program. Based on relative stability and on a number of structural features and correlations, we have devised criteria to identify structures which are unlikely to be realized as zeolites. The presence among the hypothetical nets of all 18 known uninodal zeolite structures provides a useful control set to guide our selection of filtering criteria.
Computer simulation provides the method by which each structure is evaluated and characterized. We have used accurate interatomic potential methods to calculate the relative lattice energies of these structures. This method of predicting the lattice energy was found to be effective with ionic
compounds[42] as well as
silicates,[4345] and the results were shown to agree well with experimental thermochemical
data.[46]

[1] 
Delgado Friedrichs, O., Dress, A. W. M., Huson, D. H., Klinowski, J. & Mackay, A. L.
Systematic enumeration of crystalline networks.
Nature 400, 644647 (1999).

[2] 
Meier, W. M., Olson, D. H. & Baerlocher, C.
Atlas of Zeolite Structure Types
(updates on http://www.izasc.ethz.ch/IZASC/Atlas/AtlasHome.htmll)
(Elsevier, London, 1996).

[3] 
Klinowski, J.
Hypothetical molecular sieve frameworks.
Current Opinion in Solid State & Materials Science 3, 7985 (1998).

[4] 
Wells, A. F.
ThreeDimensional Nets and Polyhedra
(Wiley, New York, 1977).

[5] 
Wells, A. F.
Further Studies of ThreeDimensional Nets,
American Crystallographic Association Monograph No. 8
(Polycrystal Book Service, Pittsburgh, 1979).

[6] 
Wells, A. F.
Structural Inorganic Chemistry
(Oxford University Press, Oxford, 1984).

[7] 
Smith, J. V.
Enumeration of 4connected 3dimensional nets and classification of framework silicates. I. Perpendicular linkage from simple hexagonal set.
Am. Mineral. 62, 703709 (1977).

[8] 
Smith, J. V.
Enumeration of 4connected 3dimensional nets and classification of framework silicates. II. Perpendicular and nearperpendicular linkages from 4.82, 3.122 and 4.6.12 nets.
Am. Mineral. 63, 960969 (1978).

[9] 
Smith, J. V.
Enumeration of 4connected 3dimensional nets and classification of framework silicates. III. Combination of helix, and zigzag, crankshaft and saw chains with simple 2D nets.
Am. Mineral. 64, 551562 (1979).

[10] 
Smith, J. V. & Bennett, J. M.
Enumeration of 4connected 3dimensional nets and classification of framework silicates: the infinite set of ABC6 nets; the Archimedean and srelated nets.
Am. Mineral. 66, 777788 (1981).

[11] 
Smith, J. V.
Enumeration of 4connected 3dimensional nets and classification of framework silicates: Combination of 41 chain and 2D nets.
Z. Kristallogr. 165, 191198 (1983).

[12] 
Smith, J. V. & Dytrych, W. J.
Nets with channels of unlimited diameter.
Nature 309, 607608 (1984).

[13] 
Smith, J. V. & Bennett, J. M.
Enumeration of 4connected 3dimensional nets and classification of framework silicates: Linkages from the two (52.8)2(5.82)1 2D nets.
Am. Mineral. 69, 104111 (1984).

[14] 
Bennett, J. M. & Smith, J. V.
Enumeration of 4connected 3dimensional nets and classification of framework silicates. 3D nets based on the 4.6.12 and (4.6.10)4 (6.6.10): 2D nets.
Z. Kristallogr. 171, 6568 (1985).

[15] 
Hawthorne, F. C. & Smith, J. V.
Enumeration of 4connected 3dimensional nets and classification of framework silicates  bodycentered cubic nets based on the rhombicuboctahedron.
Can. Miner. 24, 643648 (1986).

[16] 
Hawthorne, F. C. & Smith, J. V.
Enumeration of 4connected 3dimensional nets and classification of framework silicates. 3D nets based on insertion of 2connected vertices into 3connected plane nets.
Z. Kristallogr. 175, 1530 (1986).

[17] 
Smith, J. V. & Dytrych, W. J.
Enumeration of 4connected 3dimensional nets and classification of framework silicates. 3D nets based on doublebifurcated chains and the 4.82 and 4.6.12 3connected plane nets.
Z. Kristallogr. 175, 3136 (1986).

[18] 
Hawthorne, F. C. & Smith, J. V.
Enumeration of 4connected 3dimensional nets and classification of framework silicates. Combination of zigzag and saw chains with 63, 3.122, 4.82, 4.6.12 and (52.8)2(5.82)1 nets.
Z. Kristallogr. 183, 213231 (1988).

[19] 
Smith, J. V.
Topochemistry of zeolites and related materials. 1. Topology and geometry.
Chem. Rev. 88, 149182 (1988).

[20] 
Richardson, J. W., Smith, J. V. & Pluth, J. J.
Theoretical nets with 18ring channels: Enumeration, geometrical modeling, and neutron diffraction study of AlPO454.
J. Phys. Chem. 93, 82128219 (1989).

[21] 
Smith, J. V.
Topology of nets and stereochemistry of molecular sieves.
ACS Abstracts 205, 157IEC (1993).

[22] 
Andries, K. J. & Smith, J. V.
Opening, stellation and handle replacement of edges of the cube, tetrahedron and hexagonal prism. Application to 3connected 3dimensional polyhedra and (2,3)connected polyhedral units in 3dimensional nets.
Proc. Royal Soc. London A444, 217238 (1994).

[23] 
Andries, K. J. & Smith, J. V.
(4;2)Connected threedimensional nets related to the mixedcoordinated framework structures AlPO415, AlPO4CJ2 and AlPO412.
Acta Cryst. A50, 317325 (1994).

[24] 
Han, S. X. & Smith, J. V.
Fourconnected threedimensional nets generated from the 1,3stellated cube: Topological analysis and distanceleastsquares refinement.
Acta Cryst. A50, 302307 (1994).

[25] 
Alberti, A.
Possible 4connected frameworks with 441 unit found in heulandite, stilbite, brewsterite, and scapolite.
Am. Mineral. 64, 11881198 (1979).

[26] 
Sato, M.
Derivation of possible framework structures formed from parallel four and eightmembered rings.
Acta Cryst. A35, 547553 (1979).

[27] 
Sato, M.
Z. Kristallogr. 161, 187 (1982).

[28] 
Sato, M.
in 6th International Zeolite Conference (eds. Olson, D. H. & Bisio, A.) (Butterworths, Guildford, U.K., Reno, U.S.A., 1015 July 1983, 1984).

[29] 
Sato, M.
Framework topology of tectosilicates and its characterization in terms of coordination degree sequence.
J. Phys. Chem. 91, 46754681 (1987).

[30] 
Sherman, J. D. & Bennett, J. M.
ACS Adv. Chem. Ser. 121, 52 (1973).

[31] 
Barrer, R. M. & Villiger, H.
The crystal structure of the synthetic zeolite L.
Z. Kristallogr. 128, 352370 (1969).

[32] 
O'Keeffe, M.
Dense and rare fourconnected nets.
Z. Kristallogr. 196, 2137 (1991).

[33] 
O'Keeffe, M. & Brese, N. E.
Uninodal 4connected 3D nets. I. Nets without 3rings or 4rings.
Acta Cryst. A48, 663669 (1992).

[34] 
O'Keeffe, M.
Uninodal 4connected 3D nets. II. Nets with 3rings.
Acta Cryst. A48, 670673 (1992).

[35] 
O'Keeffe, M.
Uninodal 4connected 3D nets. III. Nets with three or four 4rings at a vertex.
Acta Cryst. A51, 916920 (1995).

[36] 
O'Keeffe, M. & Hyde, S. T.
The asymptotic behavior of coordination sequences for the 4connected nets of zeolites and related structures.
Z. Kristallogr. 211, 7378 (1996).

[37] 
O'Keeffe, M. & Hyde, B. G.
Crystal Structures I: Patterns and Symmetry (Mineralogical Association of America Monograph, Washington, D.C., 1996).

[38] 
Boisen, M. B., Gibbs, G. V., O'Keeffe, M. & Bartelmehs, K. L.
A generation of framework structures for the tectosilicates using a molecularbased potential energy function and simulated annealing strategies.
Microporous Mesoporous Mat. 29, 219266 (1999).

[39] 
Akporiaye, D. E. & Price, G. D.
Systematic enumeration of zeolite frameworks.
Zeolites 9, 2332 (1989).

[40] 
Treacy, M. M. J., Randall, K. H., Rao, S., Perry, J. A. & Chadi, D. J.
Enumeration of periodic tetrahedral frameworks.
Z. Kristallogr. 212, 768791 (1997).

[41] 
Draznieks, C. M., Newsam, J. M., Gorman, A. M., Freeman, C. M. & Férey, G.
De novo prediction of inorganic structures developed through automated assembly of secondary building units (AASBU method).
Angew. Chem. Int. Ed. 39, 22702275 (2000).

[42] 
Catlow, C. R. A., Dixon, M. & Mackrodt, W. C.
Computer Simulation of Solids.
Lecture Notes in Physics, SpringerVerlag, Berlin 166, 130161 (1982).

[43] 
Catlow, C. R. A., Thomas, J. M., Parker, S. C. & Jefferson, D. A.
Simulating silicate structures and the structural chemistry of pyroxenoids.
Nature 295, 658662 (1982).

[44] 
Sanders, M. J., Leslie, M. & Catlow, C. R. A.
Interatomic potentials for SiO2.
J. Chem. Soc., Chem. Commun., 12711273 (1984).

[45] 
Parker, S. C., Catlow, C. R. A. & Cormack, A. N.
Structure prediction of silicate minerals using energy minimization techniques.
Acta Cryst. B40, 200208 (1984).

[46] 
Henson, N. J., Cheetham, A. K. & Gale, J. D.
Theoretical calculations on silica frameworks and their correlation with experiment.
Chem. Mater. 6, 16471650 (1994). 
