Cluster functions

To build up vin the cluster function (1) we use the functions (PvA vA2---fvBi 9vs2 -- all of which satisfy the strong orthogonality condition in the sense of to (2), but do not satisfy the strong orthogonality needed for (1) We therefore consider the linear combination... 

Bn is the cluster function defined previously and the last equation defines the cluster sum 2 (n). For species in the set Nx it is found ... 

In writing explicit results an extension of the notation is useful, is the /(2) cluster function for defects number i and j... 

The problem at hand is the evaluation of the activity coefficient defined in Eq. (76). It will be assumed that only pairwise interactions between the defects need be considered at the low defect concentrations we have in mind. (The theory can be extended to include non-pairwise forces.23) Then the cluster function R(n) previously defined in Eq. (78) is the sum of all multiply connected diagrams, in which each bond represents an /-function, which can be drawn among the set of n vertices, the /-function being defined by Eqs. (66), (56), and (43). The Helmholtz free energy of interaction of two defects appearing in this definition can be written as... 

One such systematic generalization was obtained by Cohen,8 whose method is now given the point of departure was the expansion in clusters of the non-equilibrium distribution functions. This procedure is formally analogous to the series expansion in the activity where the integrals of the Ursell cluster functions at equilibrium appear in the coefficients. Cohen then obtained two expressions in which the distribution functions of one and two particles are given in terms of the solution of the Liouville equation for one particle. The elimination of this quantity between these two expressions is a problem which presents a very full formal analogy with the elimination (at equilibrium) of the activity between the Mayer equation for the concentration and the series... 

These expressions are analogous to the series expansions of the equilibrium distribution functions in terms of the activity in which appear, in the coefficients, the integrals of the Ursell cluster functions Us (see, for example, ref. 30). 

It is now clear that in addition to their widespread involvement in electron transfer pathways, iron-sulfur clusters function as catalytic centers in a wide variety of enzymes. The first example of such an enzyme is aconitase. It was at first thought that the role of the iron-sulfur group was regulatory, but it is now clear that in this enzyme the iron-sulfur group is part of the catalytic site. One of the iron atoms can coordinate water or hydroxyl and plays a key role in the isomerization catalyzed by the enzyme (Emptage et al., 1983). 

Figure 6.9 Schematic representation of a gold nanoparticle containing an average of 285 Au atoms in the core and 8 iron-cluster-functionalized thiolates.49...

Two [Fe4S4] clusters have been identified and analyzed in bacterial and archaeal adenylylsulfate reductases. These enzymes are of importance in the sulfur cycle. The role of the iron-sulfur centres in relation to the third cofactor, FAD, has been studied. It was shown that the [Fe4S4] clusters function were electron transport guiding two electrons to the FAD catalytic site.234 A novel [Fe4S4] cluster with a high spin ground state (S = 3/2) was observed in the catalytic site of E. coli nitrate reductase A.235... 

A similar equation may be written to describe the evolution of the singlet density of the reactant C. Cluster functions are introduced and, after using a super-position approximation again, the analysis follows that of Kapral [285] very closely with the complication that, instead of only singlet XA and doublet xAB correlation functions, it is necessary to consider the equation for C in tandem. This requires the use of the matrix notation for compactness. 

Using cluster functions and a super-position approximation which ignores the four-body correlation function x BSS(1234, f it jg possible to write simultaneous equations for the doublet and triplet correlation functions in a manner analogous to the homogeneous reaction case [286]. 

The dipole moment induced in a molecule, or in a group of molecules, is a finite range function of the intermolecular separations, R, which falls off faster than R-3 for R —> oo. Van Kranendonk has argued that, therefore, it is possible to expand the above equation in a series of cluster functions [400, 402]. If ft( 1 n) designates the dipole moment induced in the cluster of molecules 1 n when these are present alone in the given volume V, cluster functions 1/(1 n) can be defined according to... 

The role of the iron-sulfur clusters in many of the proteins that we have just considered is primarily one of single-electron transfer. The Fe-S cluster is a place for an electron to rest while waiting for a chance to react. There may sometimes be an associated proton pumping action. In a second group of enzymes, exemplified by aconitase (Fig. 13-4), an iron atom of a cluster functions as a Lewis acid in facilitating removal of an -OF group in an a,P dehydration of a carboxylic acid (Chapter 13). A substantial number of other bacterial dehydratases as well as an important plant dihydroxyacid dehydratase also apparently use Fe-S clusters in a catalytic fashion.317 Fumarases A and B from E. coli,317 L-serine dehydratase of a Pepto-streptococcus species,317-319 and the dihydroxyacid... 

The summation extends over all pairs of segments in the solution. We may specify (r) in terms of the cluster function p(r) defined as... 

This analysis can be extended to many-body interactions. As an example we consider the first order contribution of the three-body cluster function... 

Taking into account the results reported here on the various fractions, Figure 3 shows structures consistent with the data. Molecular formulae were determined from the elemental analyses of the fractions. Other functionalities and linking units could also satisfy the data however, those shown were considered most likely based on the results and steric and stability considerations. The structures in Figure 3 contain only major features such as the nature of the aromatic clusters, as well as the approximate numbers of aromatic and naphthenic rings per cluster. Functional groups and side chains are indicated only in a qualitative manner. 

Arrigo etcd., 1991 Clustering Function 2L/Kohonen Reall/Map(30)... 

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