Analysis of voids in crystal structures: the methods of `dual' crystal chemistry

Abstract

The theoretical basics of the analysis of voids in crystal structures by means of Voronoi±Dirichlet polyhedra (VDP) and of the graph theory are stated. Topological relations are considered between VDPs and atomic domains in a crystal ®eld. These relations allow the separation of two non-intersecting topological subspaces in a crystal structure, whose connectednesses are de®ned by two ®nite`reduced' graphs. The ®rst,`direct', subspace includes the atoms (VDP centres) and the network of interatomic bonds (VDP faces), the second, dual', one comprises the void centres (VDP vertices) and the system of channels (VDP edges) between them. Computer methods of geometrical±topological analysis of the`dual' subspace are developed and implemented within the program package TOPOS. They are designed for automatically restoring the system of channels, visualizing and sizing voids and void conglomerates, dimensional analysis of continuous void systems, and comparative topological analysis of`dual' subspaces for various substances. The methods of analysis of dual' and`direct' subspaces are noted to differ from each other only in some details that allows the term`dual' crystal chemistry to be introduced. The ef®ciency of the methods is shown with the analysis of compounds of different chemical nature:

Dilithium zirconium hexafluoride Li2ZrF6 at high pressures: A new monoclinic phase

“…Other parameters independent of the coordination number of the zirconium atoms are: (1) the radius of the spherical domain that corresponds to the sphere of volume V VDP , (2) the total area of the surface of the VD polyhedron, and (3) the dimensionless second moment of inertia of the VD polyhedra, which is a parameter characterizing the uniformity of the fluorine arrangement around the Zr atoms. In Table 2, we give the V VDP , R sd , S VDP , and G 3 parameters [20] for the structures of Li 2 ZrF 6 at atmospheric conditions (P 31m, ZZ1) [2], synthesized at 11 GPa and 1063 K (P2 1 /c, ZZ4) [11], and above 10 GPa at room temperature (C2/c, ZZ4) ( Table 1 and Fig. 5).…”

“…Hence, to ensure a proper distribution of all the cations [21], it was assumed in the search for the lithium atoms that they would occupy the general sites 8f with the positional parameters x and y close to 0.25 and 0.5, respectively. Subsequently, a set of four voids, for which the hypothetical Li-Li and Li-Zr distances in the cationic sublattice as well as the hypothetical Li-F bond lengths were crystallographically acceptable [20,21], turned out to be the ones with the octahedral coordination to the F atoms. The respective x, y, and z positional parameters of these voids were then averaged to give one position of the lithium atoms in the structural model for Li 2 ZrF 6 (C2/c, ZZ4) at 14.8 GPa.…”

“…The fluorine atoms are distributed in three nonequivalent general 8f sites (x,y,z). A topological analysis of the globally optimized ZrF K2 6 sublattice was carried out using the program DIRICHLET of the TOPOS package [20]. Voronoi-Dirichlet polyhedra (VDP) for the first coordination spheres of the fluorine sublattice were constructed to determine its geometric and topological characteristics.…”

Computational characterization and prediction of metal–organic framework properties

“…[90] These grid-based methods are computationally expensive, in particular for high-accuracy determinations which require very fine mesh spacing. An alternative to the use of gridbased sampling to characterize pore accessibility exists, based on Voronoi decomposition (depicted on Figure 8), [91,92] which for a given arrangement of atoms in a periodic domain provides a graph representation of the void space. The analysis of the Voronoi network and the accessibility of its nodes can then yield information into the components of the pore system, the dimensionality of the different channel systems, and the associated surface area and volume.…”

The relation between atomic shape and crystal structure illustrated by metallic elements

“…e) The shape of the Wigner-Seitz-cell in the tungsten structure type can be better explained and its interpretation becomes free of contradictions. According to Blatov [17] the faces of the Wigner-Seitz-cell (in the case of a monoatomic basis) resp. the faces of the Voronoi-Dirichlet polyhedron (in the case of a basis with more than one atom) can be assigned to bonds to surrounding atoms and the vertices can be assigned to interstices.…”