Definition of a cluster

Definitions are useful if they permit a body of information to be organized around a model. The more information organized the better the model. But a caution must be 

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MC simulations are performed with N = 10 molecules, each with four n.n. molecules on a 2d square lattice, at constant P and T, and with the same model parameters as for the MF analysis. To each molecules we associate a cell on a square lattice. The Wolffs algorithm is based on the definition of a cluster of variables chosen in such a way to be thermodynamically correlated." To define the Wolffs cluster, a bond index (arm) of a molecule is randomly selected this is the initial element of a stack. The cluster is grown by first checking the remaining arms of the same initial molecule if they are in the same Potts state, then they are added to the stack with probability Psame = ttiin... 

Al the basis of the polyclustcr model, there lies the assumption of the presence of one or several types of atom LO in a solid. At this point, the polycluster model is close to the so-called stereochemical models [6.8]. Besides, it includes the definition of a cluster as a set of locally ordered atoms and the definition of the boundaries as the closing of this set. Finally, the assumption that clusters conjoin along common boundaries completes the definition of a polyclustcr structure. The polycluster model includes a rather wide set of structures. This model was suggested in [6.26, 27] and developed and applied while describing various properties of metallic glasses [6.28 33]. 

One important class of cluster catalysts are dimetallic catalysts such as Rh2(carboxylate)4, which effect additions and insertions of carbenoids derived from diazoalkanes. This class is not included below because the strict definition of a cluster as containing three or more metal atoms has been adopted. This area of homogeneous catalysis has been reviewed recently " and is treated in the current edition concerning transition metals in organic synthesis. 

A few more words about the definition of a cluster are appropriate. At the time the term was proposed, there had been occasional use of the word cage . This... 

The most fitting definition of a cluster is probably a geographic concentration of mutually interconnected companies, specialized suppliers, service providers, companies in similar fields and affiliated institutions, such as universities, agencies, and business associations of different orientations that compete but also cooperate ... 

PIsanI C, Orlando R and Cora F 1992 On the problem of a suitable definition of the cluster In embedded-cluster treatments of defects In crystals J. Chem. Phys. 97 4195-204... 

Clustering techniques are mostly based on the concept of similarity expressed through the definition of a metric (distances calculus rule) in... 

Additional types of k C-k E chelating ylide complexes merit mentioning here, in addition to the species already presented, and some of them are shown in Scheme 24. The first is formed by ylides containing a pyridine functionality as substituent of the ylidic carbon. This type of ligand has allowed the S3mthesis of many loose clusters (82) of Cu, Ag, and Au which show weak metal( / °)-metal( / °) interactions. These facts have prompted the definition of a new phenomenon numismophilicity) to account for these weak interactions, uniquely shared by the three coinage metals [164]. Nitrogen and sulfur keto-stabilized ylides have been reacted with Pt and... 

Whereas absorption spectra can be obtained at a given temperature via Monte-Carlo t q)e simulations, the reach of equilibrium in an excited state of an isolated cluster is less obvious, and even less is the definition of a relevant temperature. In any case, the final state may be strongly dependent on the excitation process. Here we will ignore the vibrations of the Na(3p)Arn cluster. We assume a Franck-Condon type approximation and that emission takes place from relaxed equilibrium geometry structures on the Na(3p)Arn excited PES. The Einstein coefficients of the lines of emission towards the ground state at energy AE are given by... 

An advantage of a cluster calculation for the lateral interactions is that one can sometimes make a model that definitely has only one type of lateral interaction. For example, if one wants to compute the pair interaction between two adsorbates, one simply puts just these two adsorbates on the cluster. With a periodic calculation this is not possible the periodicity always gives you an infinite number of adsorbates, and to get the pair interaction of interest one needs to make an assumption about the other lateral interactions that are in principle present. This means that in a periodic calculation one chooses one of... 

Definitions of a metal atom cluster compound have been given (109, 233, 241, 316). Accordingly, in this review a cluster will be considered as a compound containing at least three metal atoms connected by metal-metal interactions in a triangular or polyhedral array. 

The next two entries to Table 3 are cited for completeness. Nitrogenase is treated in Chapter 7 and CO dehydrogenase in Chapter 9. Nitrogenase contains a very complex iron-sulfur cluster that includes another metal, molybdenum or vanadium. The crystal structure of the Mo variant has been determined. There is a third variant, alternative nitrogenase [92], whose cluster apparently does not contain any heterometal. That cluster would thus be a perfect candidate for our definition of a redox-catalytic iron-sulfur cluster. Unfortunately, this third nitrogenase has thus far been characterized to a much lesser extent than the other two forms. For all nitrogenases holds that the binding of N2 to the cluster has not been established [53] therefore, formally these enzymes have not yet been positively identified as redox iron-sulfur catalysts. 

In this review, we have attempted to compile a comprehensive listing of all of the mixed-metal clusters that have been prepared. It has been necessary to adhere to a specific definition of a mixed-metal cluster, and the following two criteria have been set by us for detailed coverage in this review. 

Before a molecular simulation can be performed the cluster" relevant to nucleation must be defined [11,27-29,56,57]. The definition of the cluster is important because there must be some set of criteria that differentiates it from the rest of the... 

As the first example, we shall discuss two notions of modules often used in topological models of networks (Milo et al. 2002 Ravasz et al. 2002 Itzkovitz et al. 2003 Clauset et al. 2004 Newman 2006). The first notion is that of a cluster on the basis of the nodes within the cluster having more interactions among themselves than with nodes outside of the cluster. Clusters can be quantitatively defined on the basis of a cluster coefficient. Escherichia coli has been shown to be clustered in a hierarchical fashion (Ravasz et al. 2002). Another definition is that of a network motif (Mho et al. 2002 Shen-Orr et al. 2002 Kashtan et al. 2004 Yeger-Lotem et al. 2004). These authors... 

Although a spin-orbital formulation is conceptually simple, desirable properties such as spin-adaptation may be lost when the electronic state of interest is open shell, for example. A rigorously spin-adapted theory must include spin-free definitions of the cluster operators, T, and an appropriate (perhaps multideterminant) reference wavefunction (Refs. 39, 41, 42, 156-158). Such general coupled cluster derivations are beyond the scope of this chapter, though some of the issues associated with difficult open-shell problems are discussed in the next section. 

Definition of the lower size limit for nanoparticles is fairly subjective because it is difficult to distinguish the smallest nanoparticles from multinuclear clusters and dissolved chemical species. However, the changes in physical and chemical properties of a cluster as it converts to a nanoparticle makes the smaller end of the nanoparticle size range very interesting ... 

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