Systematic enumeration of crystalline networks

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.