Voronoi - polyhedron

Another local property of interest in the study of liquids is the Voronoi polyhedron (VP), or the Dirichlet region, defined as follows Consider a specific configuration R and a particular particle /. Let us draw all the segments /, (y = 1. /) connecting 

A two-dimensional illustration of a construction of a VP is shown in Fig. 5.23. It is clear from the definition that the region (VP)/ includes all the points in space that are nearer to R/ than to any Rj j i). Furthermore, each VP contains the center of one and only one particle. 

The concept of VP can be used to generate a few local properties t the one we shall be using is the volume of the VP, which we denote by 

FIGURE 5.23. Construction of the Voronoi polygon of particle 1 in a two-dimensional system of particles. 

One way of generating new properties from previous ones is by combination. For instance, the counting function for BE and the volume of the VP is 

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The terms Voronoi polyhedron and Delaunay simplex have English geometry school origin. The first was Rodgers [136], who started using them regarding a great fundamental impact to this field from Russian mathematicians G.F. Voronoi (1868 to 1908) and B.N. Delaunay (1890 to 1980). The term V-polyhedron was used by many mathematicians [131-134], This makes sense because similar... 

In relation to the previously reported Frank-Kasper proposal, O Keeffe (1979) suggested that coordinating atoms contribute faces to the Voronoi polyhedron around the central atom, and their contributions are weighed in proportion to the solid angle subtended by that face at the centre. 

The MD simulations on isothermal crystallization processes of a-Si are introduced. To obtain a realistic a-Si structure, the /-Si prepared at 3500 K is rapidly quenched to 500 K at a cooling rate of 10 K/s. During the cooling process the structural change is observed by the Voronoi polyhedron analysis. It is shown that the unit cell of the amorphous structure becomes similar to that of crystalline structure with decrease of temperature, although its phase is still amorphous. 

Voronoi polyhedra The Voronoi polyhedron is a geometrical construction that allows an objective assignment of volume to each atom in a structure. The volume associated with an atom can be calculated by constructing a set of planes normal to the vectors between the atom of interest and its nearest neighbors. The volume inside the polyhedron is a good approximation to the volume associated with the particular atom. 

Equation (3.06) suggests that as V/ decreases, r increases exponentially and the transport is severely curtailed. Correspondingly, the motion of the particle gets confined to the Voronoi polyhedron and as expected the motion becomes more and more oscillatory. The empirical VTF equation (3.02) may be recovered from equation (3.06) by a proper substitution of Vf. One plausible assumption is that v/ s v, - Vg s Aa(T-Tg) where Vi is the... 

Another continuous-type local property of interest in the study of liquids is the Voronoi polyhedron (VP), or the Dirichlet region, defined as follows. Consider a specific configuration RN and a particular particle i. Let us draw all the segments /y(j = 1,..., N, j i) connecting the centers of particles i and j. Let Pl be the plane perpendicular to and bisecting the line Zy. Each plane P / divides the entire space into two parts. Denote by Vy that part of space that includes the point The VP of particle i for the configuration RN is defined as the intersection of all the Vy ( = 1,..., N, j i) ... 

Note that this relation is based on the fact that the volumes of the Voronoi polyhedron (VP) of all the particles add up to build the total volume of the system. Here we have an example of an explicit dependence between V and which could have been guessed. Therefore, the partial molar volume of the 4>-species can be obtained by taking the functional derivative of V with respect to (), i.e.,... 

The Voronoi calculation can be performed on protein atoms buried at interfaces as well as inside proteins. However, the procedure has a serious limitation a Voronoi polyhedron can be drawn around an atom only if it is completely surrounded by other atoms. At interfaces, only about one-third of the atoms that contribute to the interface area B have zero accessible surface area. These atoms are located mostly at the center of the interface, which biases the F/Fq ratio in an opposite way to the gap index, which is biased toward the periphery. However, high-resolution X-ray structures usually report positions for immobilized water molecules, which are abundant at interfaces (see Section II,D). These molecules may also be used to close the polyhedra, making the evaluation of Voronoi volumes possible for atoms which are surrounded by both protein atoms and immobilized water molecules (Fig. 4). On average, there are as many such interface atoms as there are completely buried atoms. Thus, a Voronoi calculation taking into account the crystallographic water molecules applies to two-thirds of the interface atoms on average instead of only one-third and up to 90% in specific cases (Lo Conte et al., 1999). 

The Frank Kasper method (Frank Kasper, 1958) defines as domain of an atom all points which are closer to the atom than to all other atoms. We draw lines joining the central atom to the surrounding atoms and then bisect these lines with perpendicular planes. The polyhedron limited by these planes is the domain of the atom (called also Voronoi polyhedron or Wigner-Seltz cell). All atoms emerging from the faces of the domain are coordinated with the central atom and define its coordination polyhedron. 

Another quantity conveying a similar meaning is the volume of the Voronoi polyhedron, which is essentially the inverse of the local density. 

In (3.4.22), the solvation volume is viewed as consisting of two terms an average Voronoi polyhedron of the solute and a change in the volume of the solvent due to solute induced changes in the distribution of the volumes of the Voronoi poly-hedra of the solvent molecules. 

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