Metastructures: homeomorphisms between complex inorganic structures and three-dimensional nets

We propose a general approach to nets and structures in which vertices represent stereochemically significant groups or clusters of atoms, and edges represent the linkage between these groups. Vertices may be single atoms, dimers, coordination polyhedra, clusters of atoms or clusters of coordination polyhedra; edges may be single chemical bonds or sets of several chemical bonds. Thus, a single net may be the basis for a family of structures that are homeomorphic to that net. The coordination of polyhedra or units around a vertex is visualized by connecting the centres of the atoms or groups at the vertices connected with the central vertex. Thus, the concept of coordination number is extended to include coordinating groups. We name a net after a homeomorphic simple structure-type for which there is a one-to-one correspondence between the vertices of the net and specific atoms of the structure, and between the edges of the net and the chemical bonds. We term the resulting more complex structures metastructures in order to distinguish them from their corresponding simple structure-types. Individual metastructures are referred to as alpha structures similar to a particular simple type. Thus, the open complex framework of [V(5)O(9)(PO(4))(2)] composition in microporous Na(v)[((V(4+)(4-w)V(5+)(1+w))O(9))(PO(4))(2)].(PO(4))(x).(OH)(y).zH(2)O is an alpha-NbO structure based on the simple net that is homeomorphic to NbO. This approach is effective in hierarchically classifying both simple close-packed structures and very complicated microporous structures.