Higher-Order Molecular Distribution Functions

Various conditional MDFs can be introduced for instance, the singlet conditional distribution function for finding a particle at X3 given two particles at Xi and X2 is 

This function will appear in the study of solvent-induced interaction in Chapter 7. The quantity defined in (5.6.3) is the average local density of particles at R3 (with orientation H3) of a system subjected to an external field of force produced by fixing two particles at Xi and X2. The arguments for this interpretation are exactly the same as those given in section 5.2. 

Higher-order correlation functions and potentials of average forces are defined in analogy with previous definitions for pairs. For instance, for = 3, we define 

is the same as fV in section 5.4. Recent simulations have shown that (5.6.6) is indeed a poor approximation. 

Consider again the basic distribution function in the T, F, N ensemble (2.1). The specific th-order MDF is defined as the probability density of finding particles 1, 2,. .., in the configuration = Xi, X2,. ., X  

In some applications, it is more advantageous to rewrite (2.112) as an average quantity, similar to Eqs. (2.13) and (2.32), namely 

Note that we have used primed vectors for quantities that are fixed. In the second form on the rhs of (2.113), we have used the basic property of the Dirac delta function, and the fact that there are altogether NlfiN — n) terms in the sum under the integral sign. The meaning of as a probability density, or as an average density, can be obtained by direct generalization of the arguments employed for and 

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The arguments leading to (3.14) are the same as those for (3.6). The new element which enters here is the triplet distribution function. Similarly, we can write formal relations for average quantities which depend on larger numbers of particles. The result would be integrals involving successively higher order molecular distribution functions. Unfortunately, even (3.14) is rarely useful since we do not have sufficient information on p(3). 

For systems that do not obey the assumption of pairwise additivity for the potential energy, equation (3.67) becomes invalid. In a formal way, one can derive an analogous relation involving higher order molecular distribution functions. This does not seem to be useful at present. However, in many applications for mixtures, one can retain the general expression (3.55) even... 

This means that the scattering is elastic, often referred to as Rayleigh scattering. (2) Each ray entering the system is scattered only once. This assumption is essential for obtaining the required relation between the intensity of the scattered beam and the pair distribution function. If multiple scattering occurs, then such a relationship would involve higher-order molecular distribution functions. 

We recall that no assumption on the total potential energy has been introduced to obtain (3.118). In the previous sections, we found relations between some thermodynamic quantities and the pair correlation functions which were based explicitly on the assumption of pairwise additivity of the total potential energy. We recall, also, that higher-order molecular distribution functions must be introduced if higher-order potentials are not... 

Relation (3.183) is useful whenever we know the free energy of the unperturbed system and when the perturbation energy is small compared with kT. It is clear that if we take more terms in the expansion (3.179), we end up with integrals involving higher-order molecular distribution functions. Therefore, such an expansion is useful only for the cases discussed in this section. For a recent review on the application of perturbation theories to liquids see Barker and Henderson (1972). 

In addition, because of its relative simplicity, we have used the Kirkwood-Buff theory for characterization of the various ideal solutions, as well as for the study of first-order deviations from ideal solutions. Although formal theories, such as that of McMillan and Mayer (1945), exist which provide expressions for higher-order deviations from ideality, their practical usefulness is limited to first-order terms only. Higher-order terms usually involve higher-order molecular distribution functions, about which little is known. 

Higher-order molecular distribution functions can be easily defined by a simple generalization of the corresponding definitions in a one-component system. 

In Section 4.8, we considered the limiting behavior of the chemical potential as 0. We have seen that the formal appearance of the chemical potential is independent of the thermodynamic variables used to describe the system. In this section, we discuss first-order deviations from DI solutions. In fact, these nonideal cases are of foremost importance in practical applications. There exist formal statistical mechanical expressions for the higher-order deviations of DI behavior however, their practical value is questionable since they usually involve higher-order molecular distribution functions. As in the previous section, we derive all the necessary relations from the Kirkwood-Buff theory, and we will be mainly concerned with the behavior of the solute A,... 

There was, however, one important follow-up paper, by Buff and Brout (1955). The reader may have noticed that the Kirkwood-Buff paper concerns exclusively those properties of solutions that can be obtained from the grand potential by differentiation with respect to pressure or particle number. Those such as partial molar energies, entropies, heat capacities, and so forth, are completely ignored. The original KB theory is an isothermal theory. The Buff-Brout paper completes the story by extending the theory to those properties derivable by differentiation with respect to the temperature. Because these functions can involve molecular distribution functions of higher order than the second, they are not as useful as the original KB theory. Yet they do provide a coherent framework for a complete theory of solution thermodynamics and not just the isothermal part. 

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. 

This chapter summarizes the most important relations between thermodynamic quantities and molecular distribution functions (MDF). The majority of these relations apply to systems obeying the assumption of pairwise additivity for the total potential energy. We shall indicate, however, how to modify the relations when higher-order potentials are to be incorporated in the formal theory. In general, higher-order potentials bring in higher-order MDF s. Since very little is known about the analytical behavior of the latter, such relationships are rarely useful in applications. 

Positional Distribution Function and Order Parameter. In addition to orientational order, some Hquid crystals possess positional order in that a snapshot at any time reveals that there are parallel planes which possess a higher density of molecular centers than the spaces between these planes. If the normal to these planes is defined as the -axis, then a positional distribution function, can be defined, where is proportional to the... 

In some Hquid crystal phases with the positional order just described, there is additional positional order in the two directions parallel to the planes. A snapshot of the molecules at any one time reveals that the molecular centers have a higher density around points which form a two-dimensional lattice, and that these positions are the same from layer to layer. The symmetry of this lattice can be either triangular or rectangular, and again a positional distribution function, can be defined. This function can be expanded in a two-dimensional Fourier series, with the coefficients in front of the two... 

Positional Distribution Function and Order Parameter. In addition to orientational order, some liquid crystals possess positional order in that a snapshot at any time reveals that there arc parallel planes which possess a higher density of molecular centers than the spaces between these planes II the normal to these- planes is defined as the -axis, then a positional distribution function. g( ). can be defined, where gOd is proportional to the fraction of molecular centers between r and + Since yO is periodic, it can he represented as a Fourier scries (a sum uf a sinusoidal function with a periodicity equal to the distance between ihe planes and its harmonics). To represent the amount ol positional order, the coefficient in front of the fundamental term is used as the order parameter. The more Ihe molecules lend to form layers, the greater the coefficient in front of ihe fundaiucnlal sinusoidal lerm and [he greaicr the order parameter for positional order,... 

For a more comprehensive description of the molecular orientation, higher-order parameters are typically needed to add independent information for the reconstruction of a probability distribution function [14]. The ability to detect molecular orientation changes during ET events of surface-confined molecules is an extremely... 

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