Pair correlation function approximation

A set of equations (15)-(17) represents the background of the so-called second-order or pair theory. If these equations are supplemented by an approximate relation between direct and pair correlation functions the problem becomes complete. Its numerical solution provides not only the density profile but also the pair correlation functions for a nonuniform fluid [55-58]. In the majority of previous studies of inhomogeneous simple fluids, the inhomogeneous Percus-Yevick approximation (PY2) has been used. It reads... 

Far from the surface, the theory reduces to the PY theory for the bulk pair correlation functions. As we have noted above, the PY theory for bulk pair correlation functions does not provide an adequate description of the thermodynamic properties of the bulk fluid. To eliminate this deficiency, a more sophisticated approximation, e.g., the SSEMSA, should be used. 

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. 

Equation 3 is exact for fluids obeying Equations 1 and 2. However, in order to compute the density n(r) from the YBG equation one must know the relationship between density distribution and the pair correlation function of Inhomogeneous fluid. Such a relationship Is not available in general. However, an approximation introduced by Fischer and Methfessel (1.) has been shown to give fairly accurate predictions of the density... 

Local Average Density Model (LADM) of Transt)ort. In the spirit of the Flscher-Methfessel local average density model. Equation 4, for the pair correlation function of Inhomogeneous fluid, a local average density model (LADM) of transport coefficients has been proposed ( ) whereby the local value of the transport coefficient, X(r), Is approximated by... 

The interfacial pair correlation functions are difficult to compute using statistical mechanical theories, and what is usually done is to assume that they are equal to the bulk correlation function times the singlet densities (the Kirkwood superposition approximation). This can be then used to determine the singlet densities (the density and the orientational profile). Molecular dynamics computer simulations can in... 

A reasonable approximation for the pair correlation function of the j8-process may be obtained in the following way. We assume that the inelastic scattering is related to imcorrelated jumps of the different atoms. Then all interferences for the inelastic process are destructive and the inelastic form factor should be identical to that of the self-correlation function, given by Eq. 4.24. On... 

Comparing now this exact equation with equation (9.1.50) we see that the Kirkwood approximation takes only a few terms into account. Next we must express equation (9.1.59) in terms of the single particle densities C (A = 0, A, B) and pair correlation functions F (r). We see that if a rate R is infinite, the corresponding distribution function p(2) is zero and... 

There have been several attempts to treat the RPM on an analogous basis. To this end, Leote de Carvalho and Evans [281] used the GMSA, Lee and Fisher [283] used the GDH, and Weiss and Schroer [239,280,284] examined several DH-based models that approximate the direct correlation function or the pair correlation function. In some cases the results depended significantly on details of the approximations. In total, none of these studies, whatever theory used, gave evidence that Nqi may be significantly smaller than observed for simple nonionic fluids. Rather the opposite seems to be the case. From this perspective, the experimental results for some ionic systems remain a mystery. 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

This relation holds only if the rate of the process is sufficiently small as compared to k — 1 )T. The fact that in this case equation (3) holds also for k > 2, means that phonons remain almost harmonic. This allows one to use in equation (7) the pair correlation approximation Dk-1(t1, t2) k — 1) > 1 (Ti, f2), where D(t, t/) = (0 q(t)q(t ) Q) is the displacement pair correlation function. The same time pairings are neglected while they give contribution to k — 2, k — 4,. ..-phonon transitions and, therefore, result in small change of the anharmonic constants Vk. Note also that the validity of equation (3) with a non-zero value of vt(t) means the existence of anomalous correlations (bf(t)) = vitfiit, these correlations depend on time. 

LDA and HDA were interpreted to be similar to two limiting structural states of supercooled liquid water up to pressures of 0.6 GPa and down to 208 K. In this interpretation, the liquid structure at high pressure is nearly independent of temperature, and it is remarkably similar to the known structure of HDA. At a low pressure, the liquid structure approaches the structure of LDA as temperature decreases [180-182]. The hydrogen bond network in HDA is deformed strongly in a manner analogous to that found in water at high temperatures, whereas the pair correlation function of LDA is closer to that of supercooled water [183], At ambient conditions, water was suggested to be a mixture of HDA-like and LDA-like states in an approximate proportion 2 3 [184-186],... 

Several kinds of approximations are involved in the derivation of Eq. (5). First, it implies that the interactions between charges can be described by an average potential, which is a mean field type of approximation. A theory devoid of this assumption can be developed in the general framework of statistical mechanics, using pair correlation functions. One of the successful approaches, based on the... 

The equations described earlier contain two unknown functions, h(r) and c(r). Therefore, they are not closed without another equation that relates the two functions. Several approximations have been proposed for the closure relations HNC, PY, MSA, etc. [12]. The HNC closure can be obtained from the diagramatic expansion of the pair correlation functions in terms of density by discarding a set of diagrams called bridge diagrams, which have multifold integrals. It should be noted that the terms kept in the HNC closure relation still include those up to the infinite orders of the density. Alternatively, the relation has been derived from the linear response of a free energy functional to the density fluctuation created by a molecule fixed in the space within the Percus trick. The HNC closure relation reads... 

Though not discussed above, in all the studies mentioned the trial wavef unctions included pair correlation functions. J j. as prescribed by Reynolds et al. ( ). Moskowitz et al. (48.49) have shown that the product of a relatively simple multiconfiguration wavefunction with pair correlation functions can provide a rather accurate approximation to the exact wavefunction. In our calculations and in those of Hammond et al. (59) the many-electron local potential, has been obtained by allowing the REP to... 

An alternative approach under the heading of weighted density approximation (WDA) attempts to model the density dependence of the pair correlation function of inhomogeneous systems. 

In fact, (4.13) is also satisfied by the x-only limit of g, i.e. its lowest order contribution in e. In the relativistic case only this limit of the pair correlation function of the RHEG, g ikplr — Ikp), specified in Eq. (B.68), is known (within the no-pair approximation [19,102]), so that we restrict the subsequent discussion to the x-only limit. 

The investigation of the RWDA, be it in the simplest form on the basis of the x-only pair correlation function of the RHEG, demonstrated that the problems of the RLDA with the cancellation of the self-interaction and with the (related) asymptotic form of the x-only potential are only corrected in part in this approximation. On the other hand, the performance of this RWDA is definitely superior with respect to the relativistic corrections (near the nucleus). Thus further improvement might be possible if a refined relativistic pair correlation function is used. 

The results summarised here are for pure water at the temperature 25°C and the density 1.000 g cm , and are obtained by solving numerictdly the Ornstein-Zemike (OZ) equation for the pair correlation functions, using a closure that supplements the hypernet-ted chain (HNC) approximation with a bridge function. The bridge function is determined from computer simulations as described below. The numerical method for solving the OZ equation is described by Ichiye and Haymet and by Duh and Haymet. ... 

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