Direct correlation functions

The expression given in this equation is easier to analyze than the equivalent one for Pr, since it involves the momentum correlation function directly rather than the inverse of the correlation matrix. It is a simple matter now to extract information related to the decay of correlations starting from Eq. (44). We note, first of all, that the functions irs(0 can be expressed in terms of the Ar(t), and these have the property lim,., Ar(i) = 0. Let us also set... 

Instead of computing the correlation functions directly, one can take the Fourier-Laplace transforms, or spectral densities... 

The Raman excitation profile is proportional to oi/((to) 2 which means that one cannot determine the time cross-correlation function directly from the observed Raman excitation profile. The indirect route is to first build a model potential energy surface for both the ground and the excited electronic states. The overlap (ijj/jih(0) is calculated by propagating the initial wave packet ij/,) on the upper electronic state and a computed resonance Raman excitation profile is obtained using Eqs. (48) and (49). The parameters of the potential energy surfaces can then be adjusted in order to get a good fit of the experimental excitation profile. In Sec. IV we shall discuss a method for a direct inversion. Another approach which has been discussed is the use of the transform theory (38,41). 

Since CMD provides an accurate approximation to the quantum position or velocity correlation function, one approach to the general correlation function problem is to introduce these correlation functions directly into an approximate expression for the correlation function [4, 5]. 

The momentum correlation function can be calculated with CMD in two ways. The first method is to compute the centroid position correlation function and to use the usual Fourier relationships between position and velocty correlation functions to obtain the momentum correlation function. The second route is to calculate the centroid momentum correlation function and then use its Fourier relation with the momentum correlation function directly (see Ref. 5). It can be shown that the two methods are equivalent, although the latter is numericaly preferable because the former has a tendency to amplify the high-frequency noise in the transform. 

Throughout this chapter, we use very simple models to introduce a number of concepts that are frequently used in the context of the theory of fluids. Examples are the pair-correlation function, direct and indirect correlations, potential of average force, nonadditivity of the triplet correlation function, and so on. All these will be introduced again in Chapter 5. However, it is easier to grasp these concepts within the simple models. This should facilitate understanding them in more complex systems. 

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation fimction of a system with a pairwise additive potential detemrines all of its themiodynamic properties. It also detemrines the compressibility of systems witir even more complex tliree-body and higher-order interactions. The pair correlation fiinctions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally detennined correlation fiinctions. We discuss the basic relations for the correlation fiinctions in the canonical and grand canonical ensembles before considering applications to model systems. 

The singlet direct correlation function C r) is defined through the relationship C Ur) = + --/t]... 

VER in liquid O 2 is far too slow to be studied directly by nonequilibrium simulations. The force-correlation function, equation (C3.5.2), was computed from an equilibrium simulation of rigid O2. The VER rate constant given in equation (C3.5.3) is proportional to the Fourier transfonn of the force-correlation function at the Oj frequency. Fiowever, there are two significant practical difficulties. First, the Fourier transfonn, denoted [Pg.3041]

Although stratification, according to the plot in Fig. 10, occurs continuously as increases, it is accompanied by a curious structural reorganization in transverse directions (i.e., parallel to the planar substrate). A suitable measure of transverse structure is the pair correlation function defined in Eq. (62). However, for simplicity we are concerned only with the in-plane pair correlation function defined as [see Eq. (62)]... 

The integrals are over the full two-dimensional volume F. For the classical contribution to the free energy /3/d([p]) the Ramakrishnan-Yussouff functional has been used in the form recently introduced by Ebner et al. [314] which is known to reproduce accurately the phase diagram of the Lennard-Jones system in three dimensions. In the classical part of the free energy functional, as an input the Ornstein-Zernike direct correlation function for the hard disc fluid is required. For the DFT calculations reported, the accurate and convenient analytic form due to Rosenfeld [315] has been used for this quantity. 

Most integral equations are based on the Ornstein-Zernike (OZ) equation [3-5]. The idea behind the OZ equation is to divide the total correlation function h ri2) iiito a direct correlation function (DCF) c r 12) that describes the fact that molecules 1 and 2 can be directly correlated, and an indirect correlation function 7( 12), that describes the correlation of molecule 1 with the other molecules that are also correlated with molecule 2. At low densities, when only direct correlations are possible, 7(r) = 0. At higher densities, where only triplet correlations are possible, we can write... 

For hard spheres, the PY approximation yields an analytic solution [15-17]. The result for the direct correlation function is... 

The polydisperse fluid structure is characterized by the total, / (r, a, (j ), and the direct, c(r, a, (jy), correlation function, both being functions of the particle diameters. These functions are related via the OZ equation (17), which is rewritten in the form... 

The pair correlation functions can be expressed directly in terms of the computed coefficients from Eq. (61) in particular, the number-number pair distribution function gN ir) and the number-number structure factor SNN k). Thus,... 

The equilibrium theory of homogeneous fluids may be constructed by using the hierarchy of the direct correlation functions [48]. This approach has been of much utility for the development of the theory of inhomogeneous simple fluids. The hierarchy of the direct correlation functions is defined by the following relation... 

We need, however, some means to calculate the direct correlation functions F2) of a nonuniform fluid. In order to make further progress, the... 

In any relation given above, the knowledge of the total or direct pair correlation functions yields an equation for the density profile. The domain of integration in Eqs. (14)-(16) must include all the points where pQ,(r) 0. In the case of a completely impermeable surface, pQ,(r) = 0 inside the wall... 

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... 

---