Clusters general formulation

Since the pioneer work of Mayer, many methods have become available for obtaining the equilibrium properties of plasmas and electrolytes from the general formulation of statistical mechanics. Let us cite, apart from the well-known cluster expansion 22 the collective coordinates approach, the dielectric constant method (for an excellent summary of these two methods see Ref. 4), and the nodal expansion method.23... 

Clustering problems can have numerous formulations depending on the choices for data structure, similarity/distance measure, and internal clustering criterion. This section first describes a very general formulation, then it details special cases that corresponds to two popular classes of clustering algorithms partitional and hierarchical. 

In this general formulation of the hierarchical clustering problem, the internal criterion J t) is calculated recursively from all the subtrees u of t. The value e(u) is sometimes called the level of the subtree u in the dendrogram. In keeping with this interpretation, e is nonincreasing along paths from the root to the leaves. 

Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg,... 

An elegant access to larger cluster units is to make use of preformed smaller clusters as they occur, e.g., in Zintl phases [8, 9]. The phase KSi (or K4Si4) contains homoatomic Si4 tetrahedra, and thus a salt-like formulation (Na )4[Si4]" with a formal electron transfer is appropriate (Fig. 1) [10]. Such homoatomic anionic building blocks occur in binary or ternary intermetallic phases A E and A Pn j with A being an alkali and alkaline-earth metal, and E = Si-Pb and Pn = P-Bi. These phases are generally available in good quantities, and - since a few members... 

The method described above is of general validity and can be applied to transition metal clusters of arbitrary shape, size, and nucleanty. It should be noted that in the specific case of a system comprising only two interacting exchanged coupled centers, our general treatment yields the same result as that of Bencini and Gatteschi (121), which was specifically formulated for dimers. In this case, the relation between the spin-projection coefficient and the on-site spin expectation value is simply given by... 

Inorganic chemists generally are unhappy with formulations of tin in oxidation state III. The formulation here ought not to be taken too seriously. Disproportionation into Sn(IV) and Sn(II) conceivably may occur. Alternatively, SnlllO(OH) may well correspond to trapping of an electron by a cluster of tin ions. 

Ignoring the correction terms in Eq. (5.6), this argument is valid only if we switch off the interaction among different chains, i.e. it holds for an ideal solution . The generalization to mutually interacting chains follows by virtue of the Linked Cluster Theorem. In the present context this important theorem can be formulated as follows. 

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