Enumeration of periodic tetrahedral frameworks

Abstract

We describe a computer method for generating periodic 4-connected frameworks. Given the number of unique tetrahedral atoms and the crystallographic space group type, the algorithm systematically explores all combinations of connected atoms and crystallographic sites, seeking the 4-connected graphs. The resulting symmetryencoded graphs are relaxed by simulated annealing to identify the regular tetrahedral frameworks.Results are presented for one unique tetrahedral atom in each of the 230 crystallographic space group types. Over 6,400 unique 3-dimensional 4-connected uninodal graphs are found when we restrict our search to those topologies that connect to nearest-neighbour asymmetric units. In any given space group, the number of graphs can depend on the choice of asymmetric unit. About 3% of the 4-connected graphs refine to reasonable tetrahedral conformations, and many are described. There is a combinatorial explosion of graphs as the number of unique vertices is increased, a result which currently restricts this method to consideration of small numbers of unique atoms.

GASP: software for geometric simulations of flexibility in polyhedral and molecular framework structures

“…[40,41] This focus on zeolites led to the following question: in a zeolite framework, is it possible in principle for the tetrahedral geometry to be made ideal and perfect? Or are the small distortions typically seen in crystal structures in fact inevitable, given the framework topology and geometry?…”

Three-dimensional Euclidean nets from two-dimensional hyperbolic tilings: kaleidoscopic examples

“…Complementary to these endeavours is the study of 'crystal nets': triply periodic embeddings of graphs in three-dimensional Euclidean space. Interest in crystal nets stems from their fundamental relevance to condensed materials from atomic crystals (Wells, 1977;O'Keeffe & Hyde, 1996;Klee, 2004) to novel framework materials (Ockwig et al, 2005;Ö hrströ m & Larsson, 2005), including carbon polymorphs (Strong et al, 2004), zeolites (Treacy et al, 1997;International Zeolite Association, 2008), related oxide (Zou et al, 2008) and alumino-phosphate (AlPO) materials (Li et al, 2008), imidazolate (ZIF) frameworks (Banerjee et al, 2008) and metalcoordination polymeric materials (Blatov et al, 2004;Batten, 2001). The microdomains of some liquid-crystalline phases of soft materials, including amphiphile and copolymer assemblies and derivative mesoporous solids, are also characterized by crystal nets (Hyde & Schroeder, 2003).…”

Apparent Scarcity of Low-Density Polymorphs of Inorganic Solids

“…The rich polymorphism of silica, especially in its low-density forms, allows for fine-tuning of applications (e.g., gas separation membranes) by choosing the best suited polymorph. In addition, 100 000þ hypothetical silica polymorphs have been predicted theoretically as fourfold-connected networks (4CNs) [3][4][5][6][7], a large fraction of which, after accurate theoretical evaluation as silica materials, were found to have comparable energetics to experimentally prepared polymorphs [8,9]. Similarly rich polymorphism has only been further observed for a select group of other inorganic solids, for example, aluminophosphates (AlPO 4 ).…”

“…Arguably, the most thorough study of the energy landscape of any TBSlike system to date was that recently applied to BN [22] which sampled systems with up to only eight atoms per unit cell finding only two hypothetical TBS polymorphs (see above). Evidently, to date, such studies have yielded only a very small number of hypothetical TBS polymorphs with respect to those derived for silica from the enumeration of 4CNs [3][4][5][6][7]. Following this latter topological approach we here sample directly the large and diverse set of 4CNs (with up to 48 atoms per unit cell when realized as a TBS) from a variety of sources.…”