Topological structure clustering

Fig. 32 The topological structure of conformation space. Left the energetic profile. Middle the connectivity graph. Right the clustering of conformations into basins.

Percolation theory rationalizes sizes and distribution of connected black and white domains and the effects of cluster formation on macroscopic properties, for example, electric conductivity of a random composite or diffusion coefficient of a porous rock. A percolation cluster is defined by a set of connected sites of one color (e.g., white ) surrounded by percolation sites of the complementary color (i.e., black ). If p is sufficiently small, the size of any connected cluster is likely to be small compared to the size of the sample. There will be no continuously connected path between the opposite faces of the sample. On the other hand, the network should be entirely connected if p is close to 1. Therefore, at some well-defined intermediate value of p, the percolation threshold, pc, a transition occurs in the topological structure of the percolation network that transforms it from a system of disconnected white clusters to a macroscopically connected system. In an infinite lattice, the site percolation threshold is the smallest occupation probability p of sites, at which an infinite cluster of white sites emerges. 

HCN is a strong dipolar species with a gas-phase dipole moment of 3.0 D [90] that may form HB networks with a rich topological structure consisting of polymerized chains, ramified, and cyclic aggregates [91, 92]. The presence of polar domains in strongly dipolar fluids has also been investigated by several works [93-95] and the stmcture and energetic properties of the HCN monomer and clusters [96-102] have been reported. 

Delete redundant sequences to make sure that the correspondence between cluster topological structures and numerical sequences is one to one. 

In paper [126] it was shown that universality of the critical indices of the percolation system was connected directly to its fractal dimension. The self-similarity of the percolation system supposes the availability of the number of subsets having order n (n = 1, 2, 4,. ..), which in the case of the structure of amorphous polymers are identified as follows [125]. The first subset (n = 1) is a percolation cluster frame or, as was shown above, a polymer cluster network. The cluster network is immersed into the second loosely packed matrix. The third (n = 4) topological structure is defined for crosslinked polymers as a chemical bonds network. In such a treatment the critical indices P, V and t are given as follows (in three-dimensional Euclidean space) [126] ... 

Figure 7 Schematic illustration for the solvent-controlled synthesis of compounds 6—10 (with blue arrows) and solvent-induced SCSC transformation/stepwise synthesis of other MOFs (with purple arrows) from 6 DMSO. The topological structures are presented in the solvent-controlled synthesis part, and topological structure of b-HjO and only coordination environments of ZUj clusters are displayed for the compounds in the SCSC transformation part. (Reprinted with permission from Ref 26. Copyright (2012) American Chemical Society.)...

The structural features of most niobium oxychlorides known to-date are summarized in Table 6.1. The use of a combination of chloride and oxide hgands leads to compounds with unique structure types [41], characterized by a remarkable variety of cluster frameworks, ranging from discrete cluster units to chains, layers, and three-dimensional nets, some topologies of which are unprecedented in compounds containing octahedral clusters. Most of the niobium oxychlorides known to date have anisotropic structures (the exceptions are Cs2LuNb,5Cli70 and PbLusNbsClisOg). 

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