Molecular distribution functions generalized

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

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 for a system of spherical particles is written in the following two equivalent forms  

P(RN) is the basic probability density in the T, V, N ensemble. In the first form on the rhs of (2.87), we have made the distinction between fixed variables Si. S and dummy variables R RN. The latter undergo 

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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. 

Relations between thermodynamic quantities and generalized molecular distribution functions... 

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). 

In all of these models, the hydrogen bonds, or the structure of liquid water, were traditionally emphasized as the main molecular reasons for the anomalous behavior exhibited by liquid water. However, underlying this relatively ill-defined concept of structure (which was much later defined in statistical mechanical terms see Sec. 2.7) lies a more fundamental principle which can be defined in molecular terms, and which does not explicitly mention the concept of structure yet is responsible for the unusual properties of liquid water. This principle was first formulated in terms of generalized molecular distribution functions in 1973. It states that there exists a range of temperatures and pressures at which the water-water interactions produce a unique correlation between high local density and a weak binding energy. Clearly, this principle does not mention structure. As will be demonstrated in this section, it is this principle, not the structure per se, which is responsible for the unique properties of water as well as of aqueous solutions. ... 

In this section, we show that the heat capacity (here, at constant volume) is not expressible in terms of the pair distribution function. In fact, we shall see that molecular distribution functions of up to order four are required for this purpose. In Chapter 5, we discuss a different possibility of expressing the heat capacity in terms of generalized molecular distribution functions. 

Generalized Molecular Distribution Functions and the Mixture-Model Approach to Liquids... 

We do not have a simple relation between the difference in the enthalpies Hi — Hu and the generalized molecular distribution functions. Therefore, the conclusion drawn here is basically intuitive. Furthermore, from the behavior of the function K)... 

A system of nonspherical particles in two dimensions is, from the computational point of view, an intermediate case between spherical particles (in either two or three dimensions) and nonspherical particles in three dimensions. The pair potential in this case depends on three coordinates (see below), compared with six in the three-dimensional case. Some very useful information on the numerical procedure, on the problem of convergence, and so forth, can thus be gained in a system which is relatively simpler than the three-dimensional case. We shall also present some results on the generalized molecular distribution function, which thus far are available only in two dimensions, yet are of relevance to the case of real liquid water. 

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... 

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