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

Abstract

We propose a general approach to nets and structures in which vertices represent stereochemically signi®cant 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 speci®c 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

Synthesis and structure of [Ce2(H2O)3](C2O4)2.5(H3C2O3) and Ce2(C2O4)(H3C2O3)4: The latter structure presents an interesting new framework, with 2-fold interpenetration

“…6), the topology of the schematized net, shown in Fig. 6, is that of Cooperite (PtS), with Ce1 as tetrahedral representing S and Ce2 as square planar representing Pt [31][32][33][34]. This is new and unprecedented topology, never observed in inorganic or coordination polymers, also known as MOF (metal-organic framework).…”

LTN-type zeolite framework as an interpenetrating net of KFI- and SOD-type parts homeomorphic to cuprite, Cu2O

“…The diamond-type net topology goes by various names as it is shared by numerous compounds: sphalerite or zinc blende arrangement (for one of the polymorphs of ZnS), or cristobalite (for one of the polymorphs of SiO 2 ). For more examples see the review by Schindler et al (1999), where these complex inorganic structures which are homeomorphic to simple three-dimensional nets are called their metastructures.…”

“…Thus, the topology of LTN is homeomorphic (topologically equivalent) to the cuprite net ( Fig. 3c), which is based on two interpenetrating nets of the diamond type offset by 1 2 , 1 2 , 1 2 (Schindler et al, 1999) and where the O atom is tetrahedrally coordinated by the Cu atoms. In cuprite the two nets are both of the same type and have no primary bonds connecting them to each other.…”

“…The densities of LTN-kfi and LTN-sod, the two component parts of the LTN-type framework, are half as large as for the LTN compound itself (Table 1), as follows from the fact that one half of each of its tetrahedrally coordinated atoms belong to its two parts. Schindler et al (1999) discuss several instances of compounds based on nets homeomorphic to cuprite. The most interesting in our context is the structure of K 2 PdSe 10 (Kim & Kanatzidis, 1992), which is also based on two interpenetrating diamond-type nets, which are compositionally different from each other: one has the composition [Pd(Se 4 ) 2 ] 2À , the other [Pd(Se 6 ) 2 ] 2À .…”

TOPOLOGICAL ENUMERATION OF DECORATED [Cu2+ 2]N SHEETS IN HYDROXY-HYDRATED COPPER-OXYSALT MINERALS

“…All subgroups of layer symmetry P32/m1 are possible, depending on the orientation of the apical bonds in a central (Cu 2+ 6 ) octahedron. Homeomorphisms (see Schindler et al 2000) to graph 3c occur in the sheets of claringbullite, posnjakite and langite, and in the synthetic compound [Cd Cu 2+ 3 (OH) 6 (NO 3 ) 2 (H 2 O)] (Table 3).…”