Voronoi cell

Construction of two-dimensional Voronoi cells by bisecting lines that connect near neighbors. 

The Voronoi deformation density approach, is based on the partitioning of space into the Voronoi cells of each atom A, that is, the region of space that is closer to that atom than to any other atom (cf. Wigner-Seitz cells in crystals see Chapter 1 of Ref. 202). The VDD charge of an atom A is then calculated as the difference between the (numerical) integral of the electron density p of the real molecule and the superposition of atomic densities SpB of the promolecule in its Voronoi cell (Eq. [42]) ... 

Figure 21.16 (a) Generalized Voronoi cell for the four-layer woodpile structure. The region of space... 

Table 4.2 Number of faces in Voronoi cells of some two- and three-dimensional arrays.

These values indicate that spherical objects prefer a body-centred cubic lattice, since this lattice maximises the number of faces in the Voronoi cell. Similarly, where the interface is cylindrical, a (two-dimensional) hexagonal network is expected. These arrays are indeed ttiose found in practice [45]. 

The spatial aggregation of topological defects evident in Figs, 5a- la is directly related to the other prominent feature of the dense WCA liquid extensive solid-like regions, which appear as rafts of nearly hexagonal Voronoi cells. As we will show, this dramatic spatial inhomogeneity is a consequence of the defect condensation transition that produces the liquid phase. Thus, it is the key qualitative feature of liquid structure that is required to understand 2D melting. 

The Voronoi cell area distribution functions for the time-averaged 3584-particle WCA system are shown in Fig. 30. These distributions consist of a single asymmetric peak that broadens and moves outward as the density decreases. As shown in Fig. 306, these distributions exhibit an isosbestic point (a single point at which all the distributions cross) in the coexistence region (in fact, there are two isosbestic points, but the isosbestic point near 1.08 is less well defined than the one near 1.17). This suggests that the distributions in this region are really the sum of two... 

Figure 30. Voronoi cell area distribution functions for the time-averaged 3584-particle WCA system, with each curve labeled by the corresponding value of p (a) solid region (b) coexistence region (c) liquid near freezing (d) liquid away from freezing.

In the coexistence region, the Voronoi cell sidelength distributions exhibit two isosbestic points (Fig. 32b), consistent with two-phase coexistence. As before, we have separated the Voronoi cell sidelength distribution for the WCA liquid at p =0.83 into contributions from ordered and disordered regions (Fig. 33). Most of the tail in the overall distribution comes from disordered regions. This is to be expected, because... 

Figure 33. Voronoi cell sidelength distribution functions for the time-averaged 3584-particle WCA liquid at p = 0.83. The figure shows the contributions from ordered regions (dotted line), disordered regions (dashed line), and the overall distribution (solid line).

The boundary of a given cell is the region of the matrix closer to that particle than any other. These cells are the Voronoi cells assuming a random distribution of particles, the distribution functions describing the interparticle distances have been calculated for both two dimensions (continuous fibers)... 

On average, the correct shape of the Voronoi cell is spherical. Thus, the correct overall material model for the assumption of a random distribution is a collection of spherical cells of different sizes, each containing a single sphere. Strictly speaking, any overall property of the material should be obtained by summing the contributions from the different cell sizes. This summing may be carried out by application of a dispersion factor to the property value found for the cell describing the overall volume fraction (12). The results presented here were obtained for the cells that describe the overall volume fraction. The application of the spatial statistical model to take into account the effect of the variable cell size is the subject of current work. 

Measurement of topological and spatial distribution of pores with Voronoi cells and Euclidean distance mapping (Venkataraman et al., 2007). 

We choose a Voronoi cell V as a fundamental domain of D. When one hits a three-dimensional boundary of a Voronoi cell, by cutting Eg parallel to the observable space, one projects into the observable space E the dual (three-dimensional) boundary. The result appears as a tile in E. The dual boundary is defined as a convex hull of all lattice points of which Voronoi domains contain the hit boundary. Instead of cutting the six-dimensional Voronoi cells, one can define a procedure on a single projected Voronoi cell y(0) with a hierarchy of all its lower-dimensional boundaries into the orthogonal space E, Vx(0) = W, the window [6]. In case of Z)g, the window W has an outer shape of a triacontahedron see Figure 12-2 top). 

Hohenberg and Kohn s proofs, 695 Kohn-Sham equation, 703 numerical integration in, 710, 717-720 Voronoi cells, 718 universal functional, 698 variation theorem, 699-700 Different Orbitals for Different Spins, see DODS... 

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