Statistical-mechanical direct correlation function

From the many tools provided by statistical mechanics for determining the EOS [36, 173, 186-188] we consider first integral equation theories for the pair correlation function gxp(ra,rp) of spherical ions which relates the density of ion / at location rp to that of a at ra. In most theories gafi(ra,rp) enters in the form of the total correlation function hxp(rx,rp) = gxp(rx,rp) — 1. The Omstein-Zemike (OZ) equation splits up hap(rx,rp) into the direct correlation function cap(ra, rp) for pair interactions plus an indirect term that reflects these interactions mediated by all other particles y ... 

In the work of Haymet and Oxtoby the direct correlation function is approximated through statistical mechanical perturbation theory about its value for a uniform liquid of density Pq. This approach relies on the earlier work of Ramakrishnan and Youssouff who showed that such a jjerturbation approach gave rather accurate results for the equilibrium phase diagram for atomic liquids. One advantage of this approach is that the only external input to the functional 2 is the direct correlation of the liquid, which can be related to the structure factor, a quantity measurable by x-ray or neutron scattering... 

The basis of the method lies in the molecular theory which relates integrals of the statistical-mechanical direct correlation function to derivatives of the total pressure and the fugacity of each species with respect to the concentration of the species of the system (4,5,6). In equation form these are... 

Compared to the effort devoted to experimental work, theoretical studies of the partial molar volume have been very limited [61, 62]. The computer simulations for the partial molar volume were started a few years ago by several researchers, but attempts are still limited. As usual, our goal is to develop a statistical-mechanical theory for calculating the partial molar volume of peptides and proteins. The Kirkwood-Buff (K-B) theory [63] provides a general framework for evaluating thermodynamic quantities of a liquid mixture, including the partial molar volume, in term of the density pair correlation functions, or equivalently, the direct correlation functions. The RISM theory is the most reliable tool for calculating these correlation functions when the solute molecule comprises many atoms and has a complicated conformation. 

D Reference Interaction Site Model The 3D RISM [80-82, 93] is a theoretical method for modeling solution phase systems based on classical statistical mechanics. The 3D RISM equations relate 3D intermolecular solvent site—solute total correlation functions (hjr)), and direct correlation functions (c (r)) (index a corresponds to the solvent sites) [80, 82] ... 

Fortunately, in the classical statistical mechanics of fluids there is a general way out of this problem, which is based on the powerful concept of the direct correlation function [22, 148]. This key mathematical object was introduced by Omstein and Zemike [178] one hundred years ago to deal with the classical fluctuations of the fluid density near the critical point. The so-called Omstein-Zemike equation at the pair level (OZ2) relates the direct correlation function between a pair of atoms cfR ) to the total correlation function through the integral... 

The closure approximation is the fundamental statistical mechanical approximation in PRISM theory. Determining the appropriate closure depends on the form of the potentials as well as the system parameters such as temperature and pressure [6]. The standard Percus-Yevick (PY) closure has been found to work well for repulsive force potentials in small molecule and macromolecular systems. The PY closure for atomic liquids can be derived using Percus method [79, 80] of a perturbative expansion of the density functional or by Stell s [8] graph summation method. The pair and direct correlation functions in PY theory are given by... 

Integral Equation and Eield-Theoretic Approaches In addition to theories based on the direct analytical extension of the PB or DH equation, PB results are often compared with statistical-mechanical approaches based on integral equation or density functional methods. We mention only a few of the most recent theoretical developments. Among the more popular are the mean spherical approximation (MSA) and the hyper-netted chain (HNC) equation. Kjellander and Marcelja have developed an anisotropic HNC approximation that treats the double layer near a flat charged surface as a series of discrete layers.Attard, Mitchell and Ninham have used a Debye-Hiickel closure for the direct correlation function to obtain an analytical extension (in terms of elliptic integrals) to the PB equation for the planar double layer. Both of these approaches, which do not include finite volume corrections, treat the fluctuation potential in a manner similar to the MPB theory of Outhwaite. 

The molecular approach, adopted throughout this book, starts from the statistical mechanical formulation of the problem. The interaction free energies are identified as correlation functions in the probability sense. As such, there is no reason to assume that these correlations are either short-range or additive. The main difference between direct and indirect correlations is that the former depend only on the interactions between the ligands. The latter depend on the maimer in which ligands affect the partition function of the adsorbent molecule (and, in general, of the solvent as well). The argument is essentially the same as that for the difference between the intermolecular potential and the potential of the mean force in liquids. 

The present state in the theory of time-dependent processes in liquids is the following. We know which correlation functions determine the results of certain physical measurements. We also know certain general properties of these correlation functions. However, because of the mathematical complexities of the V-body problem, the direct calculation of the fulltime dependence of these functions is, in general, an extremely difficult affair. This is analogous to the theory of equilibrium properties of liquids. That is, in equilibrium statistical mechanics the equilibrium properties of a system can be found if certain multidimensional integrals involving the system s partition function are evaluated. However, the exact evaluation of these integrals is usually extremely difficult especially for liquids. 

Modem quantum-chemical methods can, in principle, provide all properties of molecular systems. The achievable accuracy for a desired property of a given molecule is limited only by the available computational resources. In practice, this leads to restrictions on the size of the system From a handful of atoms for highly correlated methods to a few hundred atoms for direct Hartree-Fock (HF), density-functional (DFT) or semiempirical methods. For these systems, one can usually afford the few evaluations of the energy and its first one or two derivatives needed for optimisation of the molecular geometry. However, neither the affordable system size nor, in particular, the affordable number of configurations is sufficient to evaluate statistical-mechanical properties of such systems with any level of confidence. This makes quantum chemistry a useful tool for every molecular property that is sufficiently determined (i) at vacuum boundary conditions and (ii) at zero Kelvin. However, all effects from finite temperature, interactions with a condensed-phase environment, time-dependence and entropy are not accounted for. 

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. 

Two theoretical programs appear to promise useful information on these difficult questions. The first is the extension of Edwards techniques for direct evaluation of the necessary two particle correlation functions, as sketched in the preceding section. The second approach is to exploit the connection to percolation theory, established in Sections 3.4 and 3.6 using classical statistical methods to characterize the allowed regions and thus the eigenstates occupying them. The drawback to such a calculation, at present, is its semiclassical nature, and Edwards methods remain the only fully quantum mechanical treatment of highly disordered systems. 

This could be achieved with rigor if the statistical mechanical theory of the liquid state were quantitatively accurate. The difficulty is that complete description of the liquid structure involves specification of not only the two-body correlations, but three-body and higher correlations as well. Some properties of the three-body correlation function g R, R2,Ri) are available experimentally from the density derivative of S Q) or g R) (Egelstaff, 1992), but the experiments in general yield a direct measure of only the pair correlation function. Therefore essentially all the existing liquid state theories invoke an assumed form for the three-particle correlation function (see, e.g., Hansen and McDonald, 1976 Egelstaff, 1992) and theoretical models. 

In another procedure, which is often apphed to the study of structure transformations in materials, the correlation function y(r) (spherical average of equation (8-6)) is theoretically determined starting from basic thermodynamic (Calm, 1965) or statistical (Lebowitz, 1982) models. This is followed by the determination of I(q) (equation (8-8)) and by the direct conparison of the modeled fimction with the experimental SAXS ciuwes. This procedure can be used in order to establish the transformation mechanisms in the studied material and verify the correcmess of the proposed models. A few examples will be described in more detail in section nanophase separation. ... 

Our method for the calculation of p(P,W) is a statistical mechanical approach known as mean-field theory (refs. 1 and 5). In this approach, the properties of the nitrogen within the graphite pore are obtained directly from the forces between the constituent molecules. The parameters of the intermolecular forces are determined by (a) ensuring that the saturation pressure and saturated liquid density of the model fluid are equal to the experimental values for nitrogen at its normal boiling point (77 K, which is the temperature at which the adsorption experiments are carried out) and (b) matching the model adsoq)tion on an isolated surface to the experimental t-curve of de Boer et al. (ref. 10). Having fixed the values for these parameters, the theory is then used to calculate model isotherms for pores of a variety of widths, which are then correlated (ref. 1) as a function of pressure and pore width to yield the individual pore isotherm p(P,W). Mean-field theory is known to become less accurate as the pore size is made very small (ref. 11) even for very small pores, however, this approach is more realistic than methods based on the Kelvin equation. 

---