Interaction site fluids formalism

In this section, we review some of the important formal results in the statistical mechanics of interaction site fluids. These results provide the basis for many of the approximate theories that will be described in Section III, and the calculation of correlation functions to describe the microscopic structure of fluids. We begin with a short review of the theory of the pair correlation function based upon cluster expansions. Although this material is featured in a number of other review articles, we have chosen to include a short account here so that the present article can be reasonably self-contained. Cluster expansion techniques have played an important part in the development of theories of interaction site fluids, and in order to fully grasp the significance of these developments, it is necessary to make contact with the results derived earlier for simple fluids. We will first describe the general cluster expansion theory for fluids, which is directly applicable to rigid nonspherical molecules by a simple addition of orientational coordinates. Next we will focus on the site-site correlation functions and describe the interaction site cluster expansion. After this, we review the calculation of thermodynamic properties from the correlation functions, and then we consider the calculation of the dielectric constant and the Kirkwood orientational correlation parameters. 

The most difficult and least satisfactory part of the theories is the calculation of the cavity distribution functions for the hard sphere interaction site fluids. If the site-site formalism is being used, then the only suitable... 

The formal treatment describing the extension of perturbation theory to polar interaction site systems and to other situations where the perturbative forces are structure-determining is available and has been applied with some success to some simple models of polar diatomics. More quantitative comparisons with computer simulations need to be made. Of course, qualitatively accurate information about the structure of polar interaction site fluids has been available for some time through solutions of the SSOZ-HNC equations. However, this approach does not seem to be useful in the context of thermodynamics. Rather little attention has been paid to polarizable molecules, although these can be treated within the context of the interaction site formalism (see, for example, Chandler and, more recently, Sprik and Klein ). Although the formal treatment of the dielectric constant within the interaction site formalism is now well established, no quantitative approximations seem to emerge from any of the theories available. 

As we discussed in Section II.B, site-site correlation functions provide a very useful formalism for describing the structure of fluids modeled with interaction site potentials. In this formalism, information equivalent to g l,2) is obtained from the set of site-site correlation functions and intramolecular correlation functions. For this reason, a great deal of effort has been put into the development of integral equation theories for these correlation functions. The seminal contribution in this area was made by Chandler and Andersen, who sought to write an integral equation of the Ornstein-Zernike form in which the set of site-site total correlation functions were related to a set of site-site direct correlation functions. Their equation has the form... 

Chandler and co-workers have developed an alternative approach to density-functional theories for molecular fluids which is based upon the interaction site formalism. The straightforward generalization of Eq. (7.1.1) to a single-component interaction site system is... 

The key difficulty in developing this approach is the treatment of the influence of intramolecular correlations upon the site density, n fr). Recall from Section III.B that in the interaction site formalism, the explicit treatment of molecular orientation effects upon the correlation functions is replaced by a coupling of intramolecular and intermolecular correlations. For an inhomogeneous simple fluid, the molecular density may be written in the... 

In this section the generalized Langevin equation (GLE) for density correlation functions for molecular liquids is derived based on the memory-function formalism and on the interaction-site representation. In contrast to the monatomic liquid case, all functions appearing in the GLE for polyatomic fluids take matrix forms. Approximation schemes are developed for the memory kernel by extending the successful frameworks for simple liquids described in Sec. 5.1. 

In this chapter we have described a theory for dynamics of polyatomic fluids based on the memory-function formalism and on the interaction-site representation of molecular liquids. Approximation schemes for memory functions appearing in the generalized Langevin equation have been developed by assuming an exponential form for memory functions and by employing the mode-coupling approach. Numerical results were presented for longitudinal current spectra of a model diatomic liquid and water, and it has been discussed how the results can be interpreted in... 

The difference between and the full renormalized potential is a well-behaved function that is evaluated numerically. The interest in the renormalization procedure is now mainly a theoretical one as formal results regarding screening and other thermodynamic parameters can be obtained this way. Results applicable to both pure one-component fluids or mixtures can be obtained. The numerical solution of integral equations, such as the SSOZ and CSL equations, for sites with charge interactions should no longer use the renormalization method but rather the method we are about to describe. 

In the previous section it was shown that Andersen s formalism can be applied to derive a highly accurate and simple relationship for the monomer fraction. In order to obtain this result the renormalized association Mayer functions were employed. The applicability of Andersen s approach to more complex systems (mixtures, multiple bonds per association site, etc.) is limited by the fact that for each case the renormalized Mayer functions must be obtained by solving a rather complex combinatorial problem. A more natural formalism for describing association interactions in one-site-associating fluids is the two-density formalism of Wertheim [22, 31]. 

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