Correlation functions excess free energy

There are several ways of obtaining functionals for nonideal systems. In most cases the free energy functional is expressed as the sum of an ideal gas term, a hard-sphere term, and a term due to attractive forces. Below, I present a scheme by which approximate expression for the free energy functional may be obtained. This approach relies on the relationship between the free energy functional and the direct correlation function. Because the direct correlation functions are defined through functional derivatives of the excess free energy functional, that is,... 

The value of DFT is evidently dependent on the accessibility and accuracy of the grand potential functional, Si [p(r)]. The usual practice is to treat the molecules as hard spheres and divide the fluid-fluid potential into attractive and repulsive parts. A mean field approximation is used to simplify the former by the elimination of correlation effects. The hard sphere term is further divided into an ideal gas component and an excess component (Lastoskie etal., 1993). The ideal component is considered to be exactly local, since this part of the Helmholtz free energy per molecule depends only on the density at a particular value of r. 

Similarly to the expressions found by Singer and Chandler [80] for the RISM/HNC equations, the KH approximation (4.f3) allows one to obtain the free energy functions in a closed analytical form avoiding the necessity of numerical coupHng parameter integration. The derivation is analogous for both RISM and 3D-RISM/KH equations [28], and is shown here in the context of the 3D approach. The excess part of the solvation chemical potential, in excess over the ideal translational term, can be related to the 3D site correlation functions by the Kirkwood s charging formula... 

For a homogeneous LJ system, the excess Helmholtz free energy and the excess chemical potential pT can be evaluated from MBWR equations of state (Johnson et al., 1993). c( r —rj) is the direct correlation function (DCF) (Hansen and McDonald, 2013) of the corresponding LJ bulk system, and it has been proven that the DCF and the total correlation function satisfy the Omstein—Zemike (OZ) equation (Hansen and McDonald, 2013)... 

The key to DDFT is to truncate this hierarchy and to express g r,F t) in terms of the density. In order to arrive at Eq. (1) one assumes that g r,F i) can be approximated by the two-point correlation function of an equilibrium system with the equilibrium density distribution /Oeq(f) = /o(r t), as illustrated in Fig. 1. This is possible because for every given interaction potential V (r) and density p r t) one can find an external potential Up r i) F) such that the equilibrium density distribution Peqir) of the system with Up(r-,) r) is equal to p(r, t). Moreover, the excess parts of the fi ee energy functionals of both systems, with and without Up(rj) r), are identical. One can therefore use a sum rale relating the integral in Eq. (4) (with g r, / t) replaced by eq(f. F)) via the first direct correlation function c (r) to the functional derivative of the excess part of the free... 

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