Ring diagrams

Fig. 9. A ring diagram and the recurrence rule, (a) A ring diagram, (b) The recurrence theorem for ring diagrams.

We could of course write down the explicit form of the general nth. order ring diagrams we prefer however to establish directly an algebraic equation for the whole series and deduce the pair correlation function from its exact solution. Indeed, it is easily verified that the nth order term is derived from the (n — l)th one by adding a loop either on the upper or on the lower line. This leads immediately to the integral equation of Fig. 9b which we now write in an analytic form. 

In order to achieve our program, we still have to discuss the non-ring diagrams, which we have completely neglected so far. We want to show that, although some of these diagrams are actually... 

The same discussion may be extended to the other non-ring diagrams and it can then be shown that the Debye-HMckel theory is exact in the case of high dilution. We shall, however, not prove this result and refer the reader to the above mentioned literature for more details. 

The analysis given here on these simple examples is readily extended to more complicated situations the conclusion is that, precisely as for the ring diagrams of Coulombic interactions, the only diagrams which have to be retained are of the general type indicated in Fig. 15, where each ion interacts independently with different solvent molecules in each intermediate state where the ions are non-interacting, the wave number is conserved and equal to k or —k. Moreover, another important feature of these graphs is that they are all of the 2f type defined in Eq. (326), for which the factorization theorem holds. 

It is very easy to find the class of diagrams which have to be retained in order to get the correct relaxation effect indeed the replacement of the screened potential (314) by the Coulomb law (154) forces us to retain all the ring diagrams in order to eliminate the long-range divergences, precisely as was done in Section V-C. However, we also now have to insert on each plasma line the interactions with the solvent, as is illustrated in Fig. 16. 

Collective Coulomb interactions. We should consider the whole class of ring diagrams generated by the simple diagrams of Fig. 23, as is illustrated in Fig. 25a. 

ACXF vl 0 MBPT ORGANIZATION ACXG vl 0 MBPT LADDER DIAGRAMS ACXH vl 0 MBPT RING DIAGRAMS... 

LITERATE MANY-BODY PERTURBATION THEORY PROGRAMMING THIRD-ORDER RING DIAGRAMS... 

A literate program for third-order many-body perturbation theory ring diagram components... 

If intermediate print out is switched on, print the title Third-order hole-particle ring diagram ... 

Evaluation of the third-order ring diagram energy components ring... 

S. Wilson, Literate many-body perturbation theory programming Third-order ring diagrams, this volume. 

Structural diagrams can be simplified by using straight lines to represent the bonds between atoms in the rings. In these ring diagrams, each corner represents a carbon atom. 

Let us now assume that the chain interacts with itself. 2(6,0 S) is the sum of the contributions of the ring diagrams (see Fig. 13.6). As these diagrams have no free ends, their renormalization is trivial the only renormalization factor is the swelling 3t0(z). In fact, these ring diagrams are characterized by the insertion of one two-leg vertex (represented by a point in Fig. 13.6) but, as was shown in Chapter 12, Section 3.3.3.4, the renormalization factor of two-leg vertices can be chosen equal to unity. 

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