Hierarchical topological analysis of crystal structures: the skeletal net concept.

Topological analysis of crystal structures faces the problem of the `correct' or the `best' assignment of bonds to atoms, which is often ambiguous. A hierarchical scheme is used where any crystal structure is described as a set of topological representations, each of which corresponds to a particular assignment of bonds encoded by a periodic net. In this set, two limiting nets are distinguished, complete and skeletal, which contain, respectively, all possible bonds and the minimal number of bonds required to keep the structure periodicity. Special attention is paid to the skeletal net since it describes the connectivity of a crystal structure in the simplest way, thus enabling one to find unobvious relations between crystalline substances of different composition and architecture. The tools for the automated hierarchical topological analysis have been implemented in the program package ToposPro. Examples, which illustrate the advantages of such analysis, are considered for a number of classes of crystalline substances: elements, intermetallics, ionic and coordination compounds, and molecular crystals. General provisions of the application of the skeletal net concept are also discussed.

Local Atomic Configurations in Intermetallic Crystals: Beyond the First Coordination Shell.

We have used a combined geometrical-topological approach to analyze 21,697 intermetallic crystal structures stored in the Inorganic Crystal Structure Database. Following a geometrical scheme of close packing of balls, we have considered the three most typical polyhedral atomic environments of the icosahedral, cuboctahedral, or twinned cuboctahedral shape as well as multi-shell (up to four shells) local atomic configurations (LACs) based on these cores in 10,657 unique crystal structure determinations. In total, half of intermetallic structures have been found to contain one of these configurations, with the icosahedral LACs being the most frequent. We have revealed that even a two-shell configuration strongly predetermines the overall connectivity (topological type) of an intermetallic crystal structure. The chemical and stoichiometric composition of the multi-shell LACs generally obeys the close-packing model: the number of atoms in the subsequent shells (Nk) varies around the value Nk = 10k2 + 2, which is valid for the same size atoms, to reach the densest packing for the kth shell. Deviations from the revealed regularities often indicate inconsistencies in the crystallographic information, unusual features of the structure, or the existence of more stable phases that can be used for the validation of experimental and modeling data.

Generating triply periodic surfaces from crystal structures: the tiling approach and its application to zeolites.

Physical properties of objects depend on topological features of the corresponding triply periodic surfaces; thus topological exploration and classification of the surfaces has practical relevance. A general method is developed for generating triply periodic surfaces from triply periodic crystal structures. A triply periodic surface is derived from the natural tiling of a crystal network by an appropriate removal of some tile faces and subsequent smoothing of the resulting facet surface. The labyrinth nets of a generated triply periodic surface are built from the natural tiling, and in turn the topological parameters of the labyrinth nets are used to determine if the surface is isomorphic to a minimal surface. This method has been applied to all known 253 zeolite frameworks and 98 triply periodic surfaces were obtained, which belong to 55 topological types. Twelve surfaces were found to be isomorphic to already known triply periodic minimal surfaces (TPMSs), while four surfaces can be treated as isomorphic to new TPMSs. A procedure has also been developed for transferring the generated surfaces to a 3D-printer-readable format.

Topology versus porosity: what can reticular chemistry tell us about free space in metal-organic frameworks?

We present the results of a comprehensive geometrical and topological analysis of 3D coordination networks in 33 790 coordination polymers. We have found relations between topological descriptors and free space of the networks, and have revealed topological types that promote high porosity of metal-organic frameworks.

Nanoporous materials with predicted zeolite topologies.

An increasing number of newly synthesized materials have been found to be previously present in databases of predicted porous materials. This has been observed not only for zeolites, but also for other inorganic materials and for MOFs. We here quantify the number of synthesized zeolites that are present in a large database of predicted zeolite structures as well as the number of other inorganic crystals and MOFs present in this same database. We find a significant number of real materials are in this predicted database of zeolite-like structures. These results suggest that many other predicted structures in this database may be suitable targets for designer materials synthesis.

Packing topology in crystals of proteins and small molecules: a comparison.

We compared the topologies of protein and small molecule crystals, which have many common features - both are molecular crystals with intermolecular interactions much weaker than intramolecular interactions. They also have different features - a considerably large fraction of the volume of protein crystals is occupied by liquid water while no room is available to other molecules in small molecule crystals. We analyzed the overall and local topology and performed multilevel topological analyses (with the software package ToposPro) of carefully selected high quality sets of protein and small molecule crystal structures. Given the suboptimal packing of protein crystals, which is due the special shape and size of proteins, it would be reasonable to expect that the topology of protein crystals is different from the topology of small molecule crystals. Surprisingly, we discovered that these two types of crystalline compounds have strikingly similar topologies. This might suggest that molecular crystal formations share symmetry rules independent of molecular dimension.

Analysis of migration paths in fast-ion conductors with Voronoi-Dirichlet partition.

In terms of the Voronoi-Dirichlet partition of the crystal space, definitions are given for such concepts as ;void', ;channel' and ;migration path' for inorganic structures with three-dimensional networks of chemical bonds. A number of criteria are proposed for selecting significant voids and migration channels for alkali cations Li+-Cs+ based on the average characteristics of the Voronoi-Dirichlet polyhedra for alkali metals in oxygen-containing compounds. A general algorithm to analyze the voids in crystal structures has been developed and implemented in the computer package TOPOS. This approach was used to predict the positions of Li+ and Na+ cations and to analyze their possible migration paths in the solid superionic materials Li3M2P3O12 (M=Sc, Fe; LIPHOS) and Na1+xZr2SixP3-xO12 (NASICON), whose framework structures consist of connected M octahedra and T tetrahedra. Using this approach we determine the most probable places for charge carriers (coordinates of alkali cations) and the dimensionality of their conducting sublattice with high accuracy. The theoretically calculated coordinates of the alkali cations in MT frameworks are found to correlate to within 0.33 A with experimental data for various phases of NASICON and LIPHOS. The proposed method of computer analysis is universal and suitable for investigating fast-ion conductors with other conducting components.