Volume of the Voronoi polyhedron

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)  

A two-dimensional illustration of the construction of a VP is shown in figure 2.12. It is clear from the definition that the region (VP) includes all the points in space that are nearer to Ri than to any Rj(j yT 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 the one we shall be using is the volume of the VP, which we denote by 

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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.,... 

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. 

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. 

How we can define Nc if ligands are situated at different distances For the first time this question was addressed by Frank and Casper [227] who defined the effective coordination number (Nc ) as the number of planes in the Voronoi polyhedron. This approach assumes that all atoms have identical volume, all planes in a polyhedron are equivalent, i.e. there are no distinctions in the character of the bonds formed by different ligands. The metal structures generally conform to these criteria [228] but application of this approach to binary inorganic and complex compounds [229,230] is not straightforward and requires some accounting for different atom sizes. 

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). 

Two space-filling models have been offered to explain the bicontinuous structure [110]. One of them is the so-called Talmon-Prager model [111-113] in which Voronoi polyhedra have been used to fill a given space. The filling is proposed in such a way that one can (i) divide the polyhedra in two types, i.e. oil and water and (ii) go on increasing the number of one type of polyhedron (oil or water), correspondingly decreasing the other, so that one droplet microemulsion can smoothly transform into the other via a state where two continuous phases coexist, but are separated. In real cases, of course, a film of surfactant molecules separates the two phases. In the model, the surfactant volume is not taken into consideration. 

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