The systematic enumeration of all possible networks of atoms in inorganic structures is a matter of considerable interest. For example, 4-connected 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 4-connected 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 4-connected 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 quasi-simple 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 three-dimensional 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 4-connected 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, X-ray 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[4-6]  on three-dimensional nets and polyhedra. Smith and collaborators,[7-24]  Alberti,[25]  Sato,[26-29]  Sherman and Bennett,[30]  Barrer and Villiger,[31]  O’Keeffe and collaborators[32-38]  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 three-dimensional 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,[43-45]  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, 644-647 (1999).