Combinatorial properties of VDP faces written as the number of vertices (edges) of a face/the number of such faces (for example {4/6} for cube and {4/6 6/8} for the Fedorov cuboctahedron).
Combinatorial-topological type of VDP is determined by graph of its edges and unambiguously characterizes its combinatorial properties. It is written as n/m-k, where n and m are the numbers of VDP faces and vertices, respectively, k is ordinal number of VDP topological type with given n and m in a library of combinatorial-topological types.
Covering coefficient (Kc) is defined as a volume ratio for a sphere, circumscribed around VDP (with the center in the central atom) and the VDP volume.
DA Vector
The shift vector for an atom from the centroid of its VDP.
Dimensionless second moment of inertia of a VDP (G3) allows one to estimate uniformity degree of an environment of atom and uniformity of the whole structure. It can be considered as the characteristic of VDP sphericity. The smaller G3 value the more regular the environment and the more spherical Voronoi-Dirichlet polyhedron.
Direct, indirect, half-direct neighbour
A direct neighbour is the X atom, corresponding to a VDP face intersecting
the segment that connects it with the central atom A. If the segment A–X intersects
VDP edge, the atom will be termed a half-direct neighbor. If the segment
A–X intersects no VDP face corresponding to it (even an edge), the atom X will
be termed an indirect neighbour (center of the segment A–X lies out of
VDP surface).
Method of intersecting spheres [Serezhkin] is the method of automatic determination of coordination number of atoms. In addition to the requirement for the existence of a common face for the VDPs of chemically bonded atoms, and the requirement that they be the “direct” neighbors, the electron-shell sizes were suggested to specify by the radii of two types, Rsd and rs, corresponding to the radii of the outer valence and inner nonvalence atomic shells and to use them to formulate an additional criterion for the existence of an interatomic bond. The rs values were taken to be equal to the Slater atomic radii. The two-sphere model postulated for the quasi-isolated atoms A and X was employed to introduce the following requirements for the existence of valence interaction A–X:
(1) Two atoms at distances greater than the sum of radii of their outer spheres are considered as chemically nonbonded, because the overlap of their shells is zero (O0).
(2) The chemical interaction of two atoms begins from the point when the outer spheres of these atoms intersect (overlap of the O1 type). On further approach, the inner atomic spheres may be involved in the intersections designated as Ii depending on the number (i) of overlaps between the spheres of different types. For example, the intersections of the O4 type correspond to the fourfold overlap of both inner and outer spheres.
(3) An increase in the number of overlaps in the order O0, O1, O2, O3, O4 occurs upon a decrease in the interatomic (internuclear) distance. The atomic approach is accompanied by the energy decrease because of the electron-density redistribution resulting either in the formation of the shared electron pairs (covalent model) or in the electron transfer from one atom to another (ionic model). The strongest chemical bonds are formed by the overlaps of type O4, while the weakest bonds, by O4. The intersections of the O4 type were suggested to be treated as van der Waals forces.
(4) The coordination number (CN) of an atom is assumed to be equal to the total number of strong chemical bonds (or intersections of the O4, O3, and O2 types) formed by this atom. The intersections corresponding to the indirect neighbors are discarded. They provide evidence that the CNs in the compounds of various composition and structure can be defined simultaneously for all atoms (metals or nonmetals) from the unified positions. As distinct from the classical method, the determination of the atomic CNs by the method of intersecting spheres requires neither the a priori knowledge of the type of interatomic interactions nor the use of various systems of crystallochemical atomic radii (not counting the use of Slater radii as parameters).
Molecular coordination numbers [Peresypkina & Blatov] - number of molecules, surrounding a molecule in the crystal structure.
Number of VDP faces (Nf) - a total number of surrounding atoms including direct, indirect, and half-direct neighbours.
Packing coefficient (Kp) is defined as a volume ratio for a sphere, inscribed into VDP (with the center in the central atom) and the VDP volume.
Partial SA for A–H faces (n)
is a ratio between the total solid angle corresponding
to A–H faces
and the total solid angle of all faces. The atom A is the VDP central atom.
Radius of spherical domain (Rsd)
Rsd is a radius of sphere whose volume is equal to the
VDP volume.
Rank of interatomic contact (RC) [Shevchenko & Serezhkin] is the number of chemical bonds in the shortest chain connecting the A and B atoms in the crystal structure (example). Then, RC=1 for all valence bonds, RC=2 for A and B atoms in the chain A–D–B. If a face of the VDP of the atom A is formed by the atom B from another molecule (chain or layer), the rank of the A…B contact is set to zero, because there is no chain of bonds connecting atoms A and B in the structure of the compound.
Rank of interatomic overlap is equal to the number of the overlaps of interior and exterior spheres, circumscribed around each atom of the considered pair. The interior sphere has usually the radius equal to Slater’s radius (rs) of corresponding atom, and the exterior one has the radius equal to that of spherical domain (Rsd). The rank of interatomic overlap can vary from 0 to 4, since the maximum number of overlaps for two pairs of spheres is equal to 4. This notion is used in AutoCN program for determination of contact type (valence, specific or van der Waals) in the crystal structure.
Solid angle of VDP face (SA) is equal to the ratio of the area of a segment, which is cut by pyramid with the VDP face in the base and in central atom in the top, from unit radius sphere, circumscribed around central atom, to the total surface area of this sphere. The total solid angle of sphere is 4p steradian.
Voronoi-Dirichlet polyhedron (VDP) of a point represents a convex polyhedron of minimum volume, containing this point, and bounded by perpendicular planes, which pass through middle points of segments, connecting this point with all other points.
References
1. V. N. Serezhkin, Yu. N. Mikhailov, and Yu. A. Buslaev // Russian Journal of Inorganic Chemistry, Vol. 42, No.12, 1997, pp. 1871.
2. E. V. Peresypkina and V. A. Blatov // Acta Cryst. (2000). B56, 1035.
3. A. P. Shevchenko and V. N. Serezhkin // Russian Journal of Physical Chemistry, Vol. 78, No. 10, 2004, 1598.