Molecular distribution functions singlet

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 now introduce the singlet molecular distribution function, which is obtained from N(S) in the limit of a very small region S. First we note that Ai(Ri, S) can also be written as... 

We now observe that relation (2.45) has the same structure as relation (2.23) but with two differences. First, (2.45) refers to a system of N— 1 instead of N particles. Second, the system of AT— 1 particles is in an external field. Hence, (2.45) is interpreted as the local density at X" of a system of N— 1 particles placed in the external field Bx. This is an example of a conditional singlet molecular distribution function which is not constant everywhere. 

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). 

In the last step on the rhs, we used the definition of the singlet molecular distribution function of a system in an external potential if... 

Consider the ordinary singlet molecular distribution function MDF) in a one-component liquid ... 

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. 

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... 

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. 

We shall rederive these relations in section 4.3.1 in connection with the singlet molecular distribution function. 

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. 

We begin, however, with the singlet orientational distribution function which is shown for the three liquid crystal phases in Fig. 6. In each phase the distribution is peaked at cos of 1 showing that the preferred molecular orientation is parallel to the director. The form of the distribution function is well represented by the relatively simple function... 

The nuclear charges and the fixed charge distribution, the so-called core, which is not affected by any change in the sr-electron distribution. The singlet ground state wave function therefore describes only the n system and is given by a single closed-shell Slater determinant Aq which is constructed from a set of 5r-molecular spin orbitals (SMO) ( la), ( 2 ). etc. 

The Cooper-Mann theory of monolayer transport was based on the model of a sharply localized interfacial region in which ellipsoidal molecules were constrained to move. The surfactant molecules were assumed to be massive compared with the solvent molecules that made up the substrate and a proportionate part of the interfacial region. It was assumed that the surfactant molecules had many collisions with solvent molecules for each collision between surfactant molecules. A Boltzmann equation for the singlet distribution function of the surfactant molecules was proposed in which the interactions between the massive surfactant molecules and the substrate molecules were included in a Fokker-Planck term that involved a friction coefficient. This two-dimensional Boltzmann equation was solved using the documented techniques of kinetic theory. Surface viscosities were then calculated as a function of the relevant molecular parameters of the surfactant and the friction coefficient. Clearly the formalism considers the effect of collisions on the momentum transport of the surfactant molecules. 

The construction of Cooper and Mann (7) for the surface viscosity includes the substrate effect by a model that represents the result of very frequent molecular collisions between the small substrate molecules and the larger molecules of the monolayer. This was done by adding a term to the Boltzmann equation for the 2D singlet distribution function that is equivalent to the friction coefficient term of the Fokker-Planck equation from which Equations 24 and 25 can be constructed. Thus a Brownian motion aspect was introduced into the kinetic theory of surface viscosity. It would be interesting to derive the collision frequency of Equation 19 using the better model (7) and observe how the T/rj variable of Equation 26 emerges. 

The thermal conductivity coefficient has been derived from Browniah motion theory by Irving and Kirkwood33 in terms of the equilibrium singlet and pair distribution functions °/a) and °/Thermal conduction under a macroscopic temperature difference involves a gradient in the mean square molecular velocity rather than in the mean molecular velocity. The steady-state radial distribution function then remains spherically symmetric except for a small correction arising from the number density variation with the temperature. As the analysis introduces no new assumptions and is somewhat lengthy, it will not be reproduced here. The resulting equation for the thermal conductivity coefficient x is... 

In order to make use of the flux expressions in Sects. 6, 7, and 8, it is necessary to have the singlet distribution function and - unless the short-range force assumption is used - the doublet distribution function as well. Virtually nothing is known about the doublet distribution function. If we knew how to make a reasonable guess of this function (possibly obtainable from molecular or Brownian dynamics), then we could estimate the contributions to the fluxes in Table 1 that involve the molecule-molecule interactions. 

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