Cluster series

We proceed with cluster series which yield the integral equations. Evidently the correlation functions presented above can be defined by their diagrammatic expansions. In particular, the blocking correlation function is the subset of graphs of h rx2), such that all paths between... 

The halogeno derivatives 56b and 56d are excellent candidates for the synthesis of other derivatives of the nido-C4B2 cluster series bearing functional groups either in the 6- or the 1,6-positions. With few exceptions, the reactions of 56d with nu-... 

In the mass spectrometric investigations of most hydrogen-bonded clusters, the protonated clusters form the dominant cluster ions as a result of a rapid proton transfer reaction (Mark and Castleman 1985). Castleman and coworkers have extensively investigated the neat expansions of methanol (Morgan and Castleman 1987, 1989 Morgan et al. 1989 Zhang et al. 1991) and observed the protonated cluster ions to form the most intense cluster series. In the case of methanol, the exoergic proton transfer reaction may be written as ... 

Li clusters up to L15 are detected under the same conditions as mentioned above. There is no indication of anomalously low stabilities for Li3 or Li5 which are missing in the gas clustering series, Li2 to Li] 5, of Kimoto and Nishida (14) Table II summarizes data on Li clusters. 

The existence of mixed metal-boron clusters constitutes one example of the validity of the isolobal principles when applied to a class of compounds. The stoichiometric and structural diversity of metallaboranes is truly striking, and the idea of isolobal fragments combined with the electron counting rules is sufficient to rationalize this diversity—this fact is impressive (67,68,69, 70, 71, 72, 73, 74). We have explored the mixing of boron and metal characters in cluster MOs in the Bs nHg-nlFeCCOla) ] nido cluster series (Fig. 5). Examples of the first four members of this series (a, b, c, and d) have been well characterized (75, 76, 77), and suggestive evidence has been published for a B2H6RU3-... 

Chandler and his co-workers have taken a somewhat different point of view in seeking to improve upon the SSOZ equation. They start from the perspective that the integral equation itself is flawed since all the tractable closures correspond to the resummation of unallowed diagrams in the interaction site cluster series for the site-site total correlation function. They have formulated an integral equation in which the direct correlation function does indeed correspond to the subset of diagrams in the interaction site cluster series in which there are no nodal circles. The key to their development is a grouping of the site-site total and direct correlation functions into four classes depending upon how the root points are intersected by s-bonds. They write... 

The next step in the theory is to make a topological reduction so that the dependence upon the f bonds is replaced by a dependence upon bonds. Lupkowski and Monson show that this topological reduction cannot be carried out unless the reference total correlation functions are decomposed in the manner described in Section III.B in connection with the development of the CSL integral equation. This result is analogous to the derivation of the CSL equation as a topological reduction of the interaction site cluster series for h. They do, however, show that there is subset of diagrams in Eq. 

It turns out that the coupling strength A is the only physical parameter that appears explicitly in the cluster series and that for A < 2 all cluster integrals converge. It is therefore possible to arrange the cluster series as a power series in A. To lowest order in A one finds... 

Beyond the MSA Cluster Series Expansion and Generalized Mean... 

The smooth change in the polymer coil dimension passing through the 6-temperature is exemplified by the data of Pritchard and Caroline (1981) shown in Fig. 6.4. Here the hydrodynamic expansion factor is plotted as a fimction of the temperature. Contrary to the predictions of the blob theory, da /dr is clearly non-zero at the 0i -temperature, irrespective of the molecular weight. Its positive value is consonant with the predictions of the cluster series... 

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