Clusters connectivity

Fig. 5.7 View of the inter-cluster connectivity in the [(Zr6B)Clii xl2+x] structure c and the zigzag chains run horizontally). The Zr-Zr bonds are emphasized, and the inner halides are omitted for clarity.

Fig. 5.15 Occupation of the inter-cluster-connecting halide site X3 with a mixture of Cl and I in crystals of CsprCIs]-CS2[ Zr6B)Cli4.6l0.4].

It is interesting to note that zirconium cluster iodides only exist in the 6-12 and 6-14 type family. Therefore, this double salt is the first member of a 6-15 type family of compounds in which iodine atoms participate in the inter-cluster connectivity. 

D Framework with Craphite-like Cluster Connectivity... 

Abbreviations CC=cluster connectivity Of=overall framework HTB=hexagonal tungsten bronze HLF = Honeycomb-like layered framework [Tl5(Tl2Cl5)][(Nb6Cli204)3(Ti3CU)2]. 

Fig. 51. Scanning electron microscope image of different stages of metalization of DNA. (a) Linear chain of separated palladium clusters connecting two gold contacts (b) magnification of (a) showing clusters with diameter > 40 nm (c) continuous coated DNA strand after one development step with a diameter larger than 40 nm. Reproduced with permission from Ref. (175). Copyright 2001, American Institute of Physics.

Figure 2.9.3 shows typical maps [31] recorded with proton spin density diffusometry in a model object fabricated based on a computer generated percolation cluster (for descriptions of the so-called percolation theory see Refs. [6, 32, 33]).The pore space model is a two-dimensional site percolation cluster sites on a square lattice were occupied with a probability p (also called porosity ). Neighboring occupied sites are thought to be connected by a pore. With increasing p, clusters of neighboring occupied sites, that is pore networks, begin to form. At a critical probability pc, the so-called percolation threshold, an infinite cluster appears. On a finite system, the infinite cluster connects opposite sides of the lattice, so that transport across the pore network becomes possible. For two-dimensional site percolation clusters on a square lattice, pc was numerically found to be 0.592746 [6]. 

At the low end of the hierarchy are the TS descriptors. This is the simplest of the four classes molecular structure is viewed only in terms of atom connectivity, not as a chemical entity, and thus no chemical information is encoded. Examples include path length descriptors [13], path or cluster connectivity indices [13,14], and number of circuits. The TC descriptors are more complex in that they encode chemical information, such as atom and bond type, in addition to encoding information about how the atoms are connected within the molecule. Examples of TC descriptors include neighborhood complexity indices [23], valence path connectivity indices [13], and electrotopological state indices [17]. The TS and TC are two-dimensional descriptors which are collectively referred to as TIs (Section 31.2.1). They are straightforward in their derivation, uncomplicated by conformational assumptions, and can be calculated very quickly and inexpensively. The 3-D descriptors encode 3-D aspects of molecular structure. At the upper end of the hierarchy are the QC descriptors, which encode electronic aspects of chemical structure. As was mentioned previously, QC descriptors may be obtained using either semiempirical or ab initio calculation methods. The latter can be prohibitive in terms of the time required for calculation, especially for large molecules. 

One of the limitations of this model is that the confinement of water molecules within clusters precludes its use within the context of water transport simulation because cluster-connective hydration structure is absent. Furthermore, water activity and contractile modulus are macroscopic based concepts whose application at the nanoscopic level is dubious. P is represented by a function borrowed from macroscopic elastic theory that contains E, and there is no microstructure-specific model for the resistance to deformation that can be applied to Nation so that one is forced to use experimental tensile moduli by default. 

Another aspect that is interesting to note concerns the dependence of the DFT gap on the orientation of the wire, indeed, for each wire size the following relation holds g[100] > g[lll] > Eg [110]. As has been pointed out in Ref. [121], this is related to the different geometrical structure of the wires in the [100], [111] and [110] directions. Indeed the [100], [111] wires appear as a collection of small clusters connected along the axis, while the [110] wires resemble a linear chain. So we expect that quantum confinement effects are much bigger in the [100], [111] wires, due to their quasi zero-dimensionality, with respect to the [110] wires. Further, the orientation anisotropy reduces with the wire width and it is expected to disappear for very large wires, where the band gap approaches that of the bulk material. 

Clearly, y encodes more relevant information (probably size) than does log Kow, which does contain a size component, but also contains hydrogen bonding and polarity/polarizability components (Dearden and Bentley, 2002). Log Kow would, however, be expected to be a better descriptor for polar chemicals. In connection with this, Gerstl and Helling (1987) commented that the ability of molecular connectivities to predict log Koc was rather limited for diverse data sets. Baker et al. (2001) included two cluster connectivity terms to improve the correlation of soil sorption of a small hydrophobic data set, yielding R2 = 0.806 and 5 = 0.302. 

It has been estimated by Manners16 that 80-90% of the total number of chains in an amylopectin molecule are part of side chain clusters, while the remaining 10-20% of chains form inter-cluster connections. Thus, substantial progress in understanding the basic structure of amylopectin has been made. Although the three-dimensional structure of amylopectin in the granule is not yet known, there is evidence that it is a two-dimensional ellipsoid.17-19... 

Many other linked polyhedra similar to molecular boranes are found for B-rich solid compounds of alkali and alkaline-earth metals. There are octahedra in Li2B6, octahedra and pentagonal bipyramids in Na3B2o, dodecahedra and bicapped square antiprisms in U3B14 and icosahedra in Na2B29. The cluster connection is particularly clear for compounds of this type. 

Figure 8.12. The two pertinent empty orbitals are of b2g symmetry (cluster and ligand antibonding) and a2g symmetry (cluster antibonding) and cannot mix. Bending the Ge-H bonds to model the doubly connected Ge9 cluster units in the dimer and trimer lowers the symmetry to D2h and these two orbitals now are of big symmetry and can mix. One is stabilized and the other destabilized as shown. The former is now available at low energy to accommodate the extra lone pair. An important consequence of this exocluster orbital mixing is that the oligomers are not viewed as delocalized clusters connected by localized bonds but single delocalized entities.

Figure 2 Polyhedral atomic frameworks of idealized (a) closo, (b) nido, and (c) arachno carboranes showing conventional numbering schemes. The lines drawn between cage atoms merely illustrate cluster connectivity, and should not necessarily be taken as bonds ...

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