Diagram summation techniques

Primary products of a complex reaction can be inferred from zero conversion extrapolation of selectivity diagrams, as first described by Schneider and Frolich (16). According to this method, the molar selectivity of each product (mol product formed per 100 mol of reactant decomposed) is plotted against the percent conversion. The validity of this method has been seriously questioned (17). In principle, this method suffers from the fact that at the very low conversions required for reliable extrapolation to zero conversion, data on yields of individual products are subject to substantial analytical uncertainty. Consequently, the calculated conversion is subject to the summation of all of the errors in the yields of all of the products, and the calculated selectivities are increasingly unreliable as the conversion decreases. However, because of the vastly improved accuracy available through the use of modern analytical techniques, the criticism of the use of this method is far less valid, and significant insight into initial product distribution can be derived. 

X-ray Diffraction. Instead of deducing solid solution formation from melting point behavior, it may be deduced from X-ray powder diagrams. Using this technique the relative configurations of (-)l-NpPhMeSiH, (-)l-NpPhMeSiF, and (+) 1 -NpPhMeSiCl were shown to be the same (44). Pure and mixed crystals have similar crystallographic constants and the intensities of diffracted X-rays from the mixed crystals appeared as the summation of those of the pure component crystals. 

The diagrammatic representation of the centroid density enhances one s ability to approximately evaluate the full perturbation series [3]. For example, one can focus on a class of diagrams with the same topological characteristics. The sum of such a class results in a compact analytical expression that includes infinite terms in the summation. A very useful technique in such cases is the renormalization of diagrams [57,58]. This procedure can be applied to the vertices to define the effective potential theory diagrammatically [3, 21-23] and, in doing so, an accurate approximation to the centroid density [3]. 

There are finally attempts to apply diagrammatic techniques of many-body perturbation theory S ), with the summation of certain diagrams to infinite order, to the correlation problem in atoms and molecules. A close relationship between this kind of approach and the independent electron-pair approximation has been demonstrated >. 

An important technique in diagrammatic perturbation theory is the summation of certain classes of diagrams to infinite order. To illustrate the technique consider our two-state system. The only diagram that contributes in second-order is... 

An ideal technique for summating these individual probabilities to obtain the overall probability of the event occurring is Fault-Tree Analysis (see section 10.6), which is in essence a logic diagram with the event at the top of the tree. 

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