Voronoi–dirichlet polyhedra in crystal chemistry: theory and applications

This review is devoted to the new three-dimensional geometrical methods of crystallochemical analysis based on the Voronoi–Dirichlet partition of crystal space. It is emphasized that the model of an atom in the crystal field as its Voronoi–Dirichlet polyhedron seems to be in many cases of interest in crystal chemistry, and satisfactorily reflects the form of an atomic domain. Moreover, the Voronoi–Dirichlet approach becomes not only a particular and very limited method of crystallochemical analysis, but also a universal model of crystal structure representation on the level of its geometrical description. The scope of the review is restricted to only crystallochemical applications of the Voronoi–Dirichlet partition. In the first part, the physical meaning of the basic parameters of atomic Voronoi–Dirichlet polyhedron is considered; some crystallochemical regularities are discussed following the Voronoi–Dirichlet model of an atom. The correctness and restrictions of the model are reviewed in detail; the main attention is drawn to its relations with the quantum-mechanical representation of an atom according to Bader's approach. The subject of the second part is the method of describing polyatomic structural units with the molecular Voronoi–Dirichlet polyhedra. It is shown that this model of molecular crystals developed in the last few years, substantially extends the scope of the Voronoi–Dirichlet method in crystal chemistry for the first time to supramolecular complexes. One of the prospective new directions in this area is the analysis of cavities in a wide range of molecular and polymeric cavitandes. The third and final part concerns the problems of computer implementations of the Voronoi–Dirichlet method in crystal chemistry. The main algorithms of constructing Voronoi–Dirichlet polyhedra are considered; the capabilities of the modern program package TOPOS are discussed where all the modern methods of the Voronoi–Dirichlet crystallochemical analysis are realized as a unit.