Free Space and Natural Tilings in Metal–Organic Frameworks

In this study, we applied the natural-tiling model to perform a systematic topological analysis of the free space in 898 highly porous metal–organic frameworks (MOFs), focusing on the classification of natural tiles as minimal closed cages in a framework. The universality of the model, in combination with a rigorous algorithm for the simplification of atomic nets, enabled us to unambiguously build natural tilings across chemically and structurally diverse crystal structures. We revealed that ligand coordination has a negligible effect on the tile volume but significantly influences the tile topological type and the geometry of tile rings. Furthermore, we introduced a stiffness scale of natural tiles based on the sphericity criterion ( G 3 ), which is also an integral parameter of tile shape. As a result, we created the Topological Types of Tiles (TTT) collection, a comprehensive database of complete and simplified forms of 2030 natural tiles arranged in 202 topological types. This data set provides a topological basis for reticular chemistry–driven design of novel porous materials. We demonstrated the universality of the tiling model by its application to 105 coordination frameworks with low porosity (0–20%). Overall, the unified formalism of the topological approach ensures algorithmic consistency and reproducibility in describing both atomic networks and their associated free space. Our work demonstrates that the automated processing of crystallographic information enables one to develop new kinds of data sets suitable for material discovery, including machine learning and artificial intelligence applications.