Diagrammatic expansion

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 first contribution to h(r) is the direct correlation function c(r) that represents the correlation between a particle of a pair with its closest neighbor separated by a distance r. The second contribution is the indirect correlation function y(r), which represents the correlation between the selected particle of the pair with the rest of the fluid constituents. The total and direct correlation functions are amenable to an analysis in terms of configurational integrals clusters of particles, known as diagrammatic expansions. Providing a brief resume of the diagrammatic approach of the liquid state theory is beyond the scope of this chapter. The reader is invited to refer to appropriate textbooks on this approach [7, 9, 18, 26]. 

The present relations differ from the KM approximation since the factor 3 is replaced by the bridge function at zero separation. This feature does not seem to be unreasonable because, from diagrammatic expansions, B (r) = B r)/3 is supposed to be accurate only at very low densities. Eq. (112) presents two advantages at high density i) it provides a closed-form expression for Bother fluids than the HS model and ii) it allows to ensure a consistent calculation of the excess chemical potential by requiring only the use of the pressure consistency condition (the Gibbs-Duhem constraint, no longer required, is nevertheless implicitly satisfied within 1%). 

All these identities are useful in the evaluation of integrals in diagrammatic expansions with respect to interaction and also in the derivation of equations of motion. 

To understand the diagrammatic approach, which is introduced below, we first determine the nonequilibrium Green s functions for uncorrelated electrons in the leads and the bridge on a Keldysh contour employing a diagrammatic expansion rather than the equation of motion [45,52,53]. For such a system, the Hamiltonian is given by ... 

In the next section, we show that the Green s functions (79) and (80) become the zeroth Green s functions in diagrammatic expansion if the Coulomb interaction is considered between the bridge electrons. 

Thus, a diagrammatic expansion in the interaction representation can be presented as follows ... 

Figure 7. Diagrammatic expansion of the correlation energy through fourth order.

H-point functions are called irreducible if their diagrammatic expansions only consist of graphs which do not split into two pieces if one internal electron or photon line is cut. 

Several interesting theoretical papers have appeared dealing with molecular dynamics and excimer formation in polymer systems. Frank and coworkers have developed a model to describe the transport of electronic excitation energy in polymer chains. The theory applies to an isolated chain with a small concentration of randomly placed chromophores, and a three-dimensional transport model was used to solve the problem which is based on a diagrammatic expansion of the transport Green function. (The Green function is related to time-dependent and photostationary depolarization and to transient and steady-state trap fluorescence.) The analysis is shown to be... 

Starting during the 19th century with the gas-kinetic theory, statistical mechanics has been very successfully developed during the 20th century. Today we have at our disposal efficient theoretical tools (diagrammatic expansion,... 

By using a Feynman-diagrammatic expansion of this it is possible [3] to... 

Already some 50 years ago, attempts have been made to estimate corrections upon the HF approximation. A concise review, including the diagrammatic expansion of the correlation energy and the Wigner lattice is e.g. given by Mahan . ... 

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