The singlet generalized molecular distribution function

In this section, we present a special case of the generalization procedure outlined above. Consider the ordinary singlet MDF  

(Si) dS is the average number of particles occupying the element of volume dS. For the present treatment, we limit our discussion to spherical molecules only. As we have already stressed in section 2.1, the quantity defined in (2.88) can be assigned two different meanings. The first follows from the first form on the rhs of (2.88), which is an average quantity in the T, V, Nensemble. The second form on the rhs of (2.88) provides the probability of finding particle 1 in the element of volume dSi. Clearly, this probability is given by N iS dSJN. 

Let us now rewrite (2.88) in a somewhat more complicated way. For each configuration RN, we define the property of the particle i as 

The property of the i-th particle, defined in (2.89), is the location of particle i, giving a configuration RN which is simply Rr This is the reason for using the letter L in the definition of the function L, (rn)  

This is the number of particles whose property L attains a value within dSi at Si, given the configuration R. The average number (here in the T, V, N ensemble) of such particles is 

Al (Si) dSi is the average number of particles occupying the element of volume rfSj. For the present treatment, we specialize our discussion to 

In this section we generalize the concept of molecular distribution to include properties other than the locations and orientations of the particles. We shall mainly focus on the singlet generalized molecular distribution function (MDF), which provides a firm basis for the so-called mixture model approach to liquids. The latter has been used extensively for complex liquids such as water and aqueous solutions. 

The general procedure of defining the generalized MDF is the following. We recall the general definition of the nth-order MDF, say in the T, V, N ensemble, which we write in the following two equivalent forms  

The generalization can be carried out on both the property and the condition imposed on it. The distinction between property and condition is arbitrary and is made for convenience only. In fact, the generalization procedure involves only one concept. This will be demonstrated, along with a few examples, in the next few sections. 

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We present here an example of complementary information on the system of waterlike particles in two dimensions, obtained by the standard Monte Carlo method. The model is the same as above, but we focus our attention mainly on the singlet generalized molecular distribution functions (Chapter 5). Figure 6.24 shows a sample of 36 waterlike particles. The molecular parameters chosen for this particular illustration are... 

In Fig. 6.26, the singlet generalized molecular distribution functions Xc K) are plotted for the three cases listed in (6.132). The most prominent feature of these curves is the shift to the left of the most probable coordina-... 

Another feature of the mode of packing of waterlike particles akin to the behavior of liquid water is demonstrated by the joint singlet generalized molecular distribution function, constructed by combining the binding energy and coordination number (Fig. 6.28). The values of K) Av... 

We start with detailed definitions of the singlet and the pair distribution functions. We then introduce the pair correlation function, a function which is the cornerstone in any molecular theory of liquids. Some of the salient features of these functions are illustrated both for one- and for multicomponent systems. Also, we introduce the concepts of the generalized molecular distribution functions. These were found useful in the application of the mixture model approach to liquid water and aqueous solutions. 

We present here a few examples of generalized molecular distribution functions MDFs (see Ben-Naim 1973a). Of particular interest is the singlet GMDF. These... 

In section 2.7, we introduced the generalized molecular distribution functions GMDFs. Of particular importance are the singlet GMDF, which may be re-interpreted as the quasi-component distribution function (QCDF). These functions were deemed very useful in the study of liquid water. They provided a firm basis for the so-called mixture model approach to liquids in general, and for liquid water in particular (see Ben-Naim 1972a, 1973a, 1974). 

The notions of molecular distribution functions (MDF) command a central role in the theory of fluids. Of foremost importance among these are the singlet and the pair distribution functions. This chapter is mainly devoted to describing and surveying the fundamental features of these two functions. At the end of the chapter, we briefly mention the general definitions of higher-order MDF s. These are rarely incorporated into actual applications, since very little is known about their properties. 

Here [5], the molecular distribution functions were calculated by linearization of a generalized form of Kirkwood s integral equation [12, 13] and the increase in surface tension was computed from the molecular theory of Buff [14, 15] for this property. The ions were taken as point charges. The singlet and pair distribution functions were evaluated for very dilute ionic solutions. Then they were introduced into the statistical mechanical formulas for surface tension. 

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