Pair correlation function long range

Hemmer P C 1964 On van der Waals theory of vapor-liquid equilibrium IV. The pair correlation function and equation of state for long-range forces J. Math. Phys. 5 75... 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

In this nonvariational approach for the first term represents the potential of the exchange-correlation hole which has long range — 1/r asymptotics. We recognize the previously introduced splitup into the screening and screening response part of Eq. (69). As discussed in the section on the atomic shell structure the correct properties of the atomic sheU structure in v arise from a steplike behavior of the functional derivative of the pair-correlation function. However the WDA pair-correlation function does not exhibit this step structure in atoms and decays too smoothly [94]. A related deficiency is that the intershell contributions to E c are overestimated. Both deficiencies arise from the fact that it is very difficult to represent the atomic shell structure in terms of the smooth function p. Substantial improvement can be obtained however from a WDA scheme dependent on atomic shell densities [92,93]. In this way the overestimated intershell contributions are much reduced. Although this orbital-depen-... 

It will be useful now to review some elementary facts regarding the structure of liquids at equilibrium. When a crystalline solid melts to form a liquid, the long range order of the crystal is destroyed. However, a residue of local order persists in the liquid state with a range of several molecular diameters. The local order characteristic of the liquid state is described in terms of a pair correlation function, g-i(R)> defined as the ratio of the average molecular density, p(R), at a distance R from an arbitrary molecule to the mean bulk density, p, of the liquid... 

Figure 2 Solute-solvent pair correlation function at T = 1.37 and p = 0.40, 0.50, and 0.60. a) Short range structure b) long range structure (note change of axes).

Clearly, the conditions above hold because the system, though open with respect to D, is not open with respect to A and C individually. The KB theory applies for any three-component system of W, A, and C without any restriction on the concentrations of A and C, i.e., when the system is open to each of its components. If this happens for an electrolyte solution, clearly the conservation of the total charge in the system will not hold, and fluctuations in A and C would lead to fluctuations in the net charge of the system. One should not interpret equations (8.49), (8.51), or (8.53), as implying anything on the preferential solvation of W, A, or C. The reason is that the condition of the conservation of the total number of A and C must impose a long-range behavior on the various pair correlation functions. This is similar to a two-component system of A and B in a closed system, where we have (see section 4.2) the conservation relations... 

The long-range behavior of the pair correlation function... 

THE LONG-RANGE BEHAVIOR OF THE PAIR CORRELATION FUNCTION 325... 

We now turn to a system of interacting particles. The long range or the asymptotic behavior of the pair correlation functions are well known for closed and open systems ... 

For charged particles, the interactions are also long-ranged, but still there exists some Rcor beyond which no direct influence of the interaction is noticeable. There are several studies of the manner that the pair correlation function decays to unity [see for example Fisher and Widom (1969), Perry and Throup (1972)]. In any case, even when g(R) is of relatively long-range, we can still find a radius Rc beyond which the pair correlation is practically unity. 

We note that /n(q) is strictly independent of q whereas/22(q), which can be expressed in terms of the angular dependent pair-correlation function, can depend on q albeit as an even function of q. In the absence of long-range orientational order2, /22(q) can be regarded as a constant independent of q (for the small values of q that are probed in light scattering). 

It is of some interest that even the presence of long-range forces, such as the Coulombic ones present in simple fused salts, does not force us to contradict the conjecture that the pair correlation function of a dense fluid is virtually equal to that of a fluid composed of the uncharged cores. Stillinger has aptly summarized the arguments in support of this contention for fused salts and we cannot afford the space to repeat these arguments here. In particular, he has shown that the extension of Fowler s approximate evaluation of liquid surface tension to molten salts of the symmetrical valence type yields ... 

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