Summation of diagrams

When the electron-phonon interaction is taken into account side by side with the impurity scattering, various phonon lines must be inserted in the diagrams of Fig. la, b, c. As a result, the summation of diagrams becomes much more complicated since the diagrams now include the electronic lines with different energies. Thus, the solution of the problem is possible only in some limiting cases. 

Formally the contribution of diagram Fig. 7.4 is given by the standard quantum mechanical second order perturbation theory term. Summation over the intermediate states, which accounts for binding, is realized with the help... 

The summation of signals is represented on a block diagram as shown in Fig. 7.13. Figure 7.13a represents the relationship ... 

It is a statistical-mechanical theory of solutions to express the solvation free energy as a functional of distribution functions. Traditionally, the theory of solutions is formulated with a diagrammatic approach [13], in which an approximation is provided in a two-step procedure. In the first step, the free energy and/or distribution function is expanded with respect to the solute-solvent interaction potential function or its related function as an infinite, perturbation series. In the second step, a renormalization scheme is applied a set of functions are defined through partial summation of the series and are employed for substitution to make the infinite series more tractable. An approximation is typically introduced by neglecting diagrams of ill character. 

The core of this procedure is a graphic summation of the output characteristics of all sources involved. The summation characteristic (output) is to be plotted in a suitable diagram (Fig. 6.226) which indicates the compliance of the circuit with intrinsic safety even in the case of a simultaneous inductive and capacitive wiring. 

The relationship between the coupled-electron pair approximation (c.e.p.a.) and the many-body perturbation theory has been discussed in detail by Ahl-richs.149 All of the methods denoted by c.e.p.a. (x) (x=0, 1, 2, 3) may be related to the summation of certain classes of diagrams in the many-body perturbation theory to infinite order. For example, c.e.p.a. (0), which is Cizek s linear approximation or Hurley s c.p.a. (0) ansatz150 is equivalent to the summation of all double-excitation linked diagrams in the perturbation series. This is also denoted d.e.m.b.p.t. (double excitation many-body perturbation theory) by some workers.151 168... 

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. 

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