Connected clusters

Identify the cavities Each connected cluster of Voronoi polyhedra vertices and edges that avoid overlap with the exclusion spheres in the system identifies a distinct cavity of available space. 

For a tetrahedron, for instance, there are 4 vertices in each of which 3 edges meet it is an example of three-connected cluster. Therefore ... 

Proton conductivities of 0.1 S cm at high excess water contents in current PEMs stem from the concerted effect of a high concentration of free protons, high liquid-like proton mobility, and a well-connected cluster network of hydrated pathways. i i i i Correspondingly, the detrimental effects of membrane dehydration are multifold. It triggers morphological transitions that have been studied recently in experiment and theory.2 .i29.i ,i62 water contents below the percolation threshold, the well-hydrated pathways cease to span the complete sample, and poorly hydrated channels control the overall transports ll Moreover, the structure of water and the molecular mechanisms of proton transport change at low water contents. 

Eq (IS) can be obtained by projecting the connected cluster form of the SchrOdinger equation on the reference configuration (I>). 

Fig. 14 Group 15 rings crystallized from ammonia or ethylenediamine solutions Fig. 15 Connected clusters for E = P, As (a) (b) E22" ...

To my knowledge, G.LemaTtre is the first one to have developed the calculation in detail (Lemaitre, 1933). He is also a pioneer in this field as he was probably the first to try to connect cluster s properties with the cosmological framework, although in retrospect his arguments were not very convincing ... 

For the linear chain, any connected cluster (/x, v) is simply a set of consecutive sites n, n + 1,... n + m. Numerical results for ansatze using connected-cluster patterns involving up to M = 4 sites, for alternate chains, and up to M = 5 sites, for regular chains, have been obtained so far. The hierarchy of patterns have been defined as... 

Numerical results for such ansatze with connected-cluster patterns involving up to M = 5 bonds have been treated so far. The hierarchy of patterns have been defined as... 

The overall structure of the supercluster has the following topology (Figure 6) where each vertex represents one of the two central Mo atoms (one inside the other relative to the sphere) in the linking (Mo)Mo5 groups holding the water H-bonded connecting clusters. 

The second model extends the surface diffusion model to include the importance of the atomic placement of atoms in the randomly packed alloy. The model considers that a continuous connected cluster of the less noble atoms must exist to maintain the selective dissolution process for more than just the few monolayers of the alloy. This percolating cluster of atoms provides a continuous active pathway for the corrosion process as well as a pathway for the electrolyte to penetrate the solid. This is expected to depend on a sharp critical composition of the less noble element, below which dealloying does not occur.54, (Corcoran)5... 

T introduces true n-particle correlation, and products like T T etc., arising out of the expansion eq.(3.2 ), generate simultaneous presence of k-particle amd m-particle correlations in a (k+m)-fold excited determinants etc. The truncation of T == T then corresponds to the pair—correlation model of Sinanoglu, while incorporating higher excited states with several disjoint pair excitations induced through the powers T - The amplitudes for T may be called linked or connected clusters for n electrons. The difficulty of a linear variation method such as Cl lies in its inability to realize the cluster expansion structure eq.(3.2),in a simple and practicable manner. 

The surface atoms of three-connect clusters have three nearest neighbors within bonding distance (Figure 2.2). Many clusters formed from p-block elements are three-connect clusters. No new bonding ideas outside of the localized two-center-two-electron bond and the eight-electron rule are required to understand most... 

In the common three-connect cluster shapes illustrated in Figure 2.3 the number of electrons that can be associated with the closed clusters is 5n, where n is the... 

Another way to count the electrons in clusters restricts the count to only those associated with cluster bonding hence, skeletal-electron pairs (sep). For three-connect clusters sep = 3/2n. The assumption that orbitals associated with cluster ligand bonding can be considered independently of orbitals associated with cluster framework bonding results in an important simplification for more complex systems. However, it is a separation that is convenient rather than rigorous. We will see... 

The polyhedral shapes pertinent to this class of clusters are the deltahedra shown in Figure 2.7. Deltahedra are polyhedra containing exclusively triangular faces. The vertex connectivities are shown on the drawings. Those for n = 4 (not shown), 6 and 12 have uniform connectivities of three, four and five hence, the cluster atom centers lie on a sphere. The others with less uniform connectivities are less spherical in shape if all edge lengths are equal. As it is a three-connect cluster, the tetrahedron... 

The diagonal lines connecting clusters of equal nuclearity reveal another important relationship - as the number of cluster electron pairs increases in a step-wise fashion the cluster opens up into a net and then a web. As was pointed out in Chapter 1, this is a recurring motif - the more electron rich, the more open the structure. One can continue. Addition of three pairs of electrons to a c/oso-borane generates an even more open shape albeit a less well-accepted nomenclature. The... 

Let us focus on Ge and begin with three-connect clusters. A compound containing [Ge4]4- tetrahedra has been prepared from Na and Ge (Figure 2.23). [Ge]- is isoelectronic with As. In the sep count it is a three-electron cluster fragment like C-H. So the cluster is analogous to P4 with six two-center Ge-Ge bonds and four external lone pairs. One can see the power of the Zintl idea ME is equivalent to E where E is a p-block element one column to the right. 

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