Graphical expansions

The power of the graphical expansion technique become apparent when we write down the first five terms in Eq. (2.1.3) as... 

Using the assumption of pairwise additive potentials and the previous definition of Zi(i), we can write down the graphical expansion for p/r. 

This is convenient because the various correlation functions may be defined as functional derivatives. Armed with a graphical interpretation of functional differentiation, we can obtain graphical expansions for the correlation functions we need. If F is a set of graphs then the n order functional derivative with respect to y is written in graphical language as... 

The most compact graphical expansion we have is for the logarithm of S, so it is more convenient to consider a distribution function based upon a functional derivative of logH. This leads naturally to the Ursell cluster... 

We are now in position to derive a second equation—one that relates C2(1,2) to /i2(l,2) and other known functions. This second equation is normally called a closure relation, and when combined with the Ornstein-Zernike equation, we have a closed system of equations to solve (two coupled equations in two unknowns /ij and C2). In principle, the graphical expansion of 2(1,2) in Eq. (2.1.29) is the exact closure relation, and if we could calculate all of the graphs we would have the exact solution. In practice, this has not been possible and all closure relations involve some type of approximation. 

To generate the Percus-Yevick (PY) closure relation, we consider the graphical expansion for 02(1 >2) obtained from Eq. (2.1.26). Each graph in this expansion occurs both with and without a/2(l, 2) bond, so we can factor out (1 -l-/2(l,2)) = e2(l,2). This gives... 

We have already derived a graphical expansion for h(l, 2), but it consists of /2-bonds containing the total interaction between molecules 1 and 2. We can transform this expansion for h l,2) into an expansion containing site-site /-bonds by considering a single /-bond ... 

Continuing this process, the graphical expansion for h y r) is given by... 

Given the graphical expansion for the site-site total correlation function h y r) it is possible to generate on Omstein-Zernike like equation. We will adopt the physical approach that the total correlation function is the sum of all possible direct site-site correlations, both intramolecular correlations via an s -bond, or intermolecular correlations via a c, (r). This defines a site-site direct correlation function c y r). From the expansion (2.2.5), it follows that at most one s -bond is connected to each circle. 

Since the s points are already connected after step (1), all pairs lacking the direct f-bonds receive the sum of no bond and a direct /g-bond in step 2. The f-connected graphs filled with Cg-bonds on 2,3,..., few,..., s points as dimer, trimer,..., oligomer,..., s-mer graphs. Using this construction, the graphical expansion for In 3 can be written as... 

If we consider the graphical expansion for p(l), we can divide the graphs into two classes if the labeled point 1 is a monomer point (that is, has no incident f-bond), then the graph is in Po(l) if 1 is an s-mer point with s 2, then the graph is in Px(l). The physical interpretation of pg is as a monomer density. 

When eq 8.3 is substituted into the cluster expansion for the distribution functions in the fluid, and simplified through cancellations in graphs by taking into account eq 8.2, the result is a formal graphical expansion for the pair distribution function in terms of renormalized hydrogen-bond /-functions. This concept of saturation at the dimer level is a key element in Wertheim s theory, discussed below. 

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