Helmholtz free energy functional, density

Actually, the various equations listed in this section are insufficient to perform the complete calculation since one would first calculate the density of H2O through eq. 8.12 or 8.14. Equation 8.14 in its turn involves the partial derivative of the Helmholtz free energy function 8.15. Moreover, the evaluation of electrostatic properties of the solvent and of the Bom functions (o, Q, Y, X involve additional equations and variables not given here for the sake of brevity (eqs. 36, 40 to 44, 49 to 52 and tables 1 to 3 in Johnson et ah, 1991). In spite of this fact, the decision to outline here briefly the HKF model rests on its paramount importance in geochemistry. Moreover, most of the listed thermodynamic parameters have an intrinsic validity that transcends the model itself... 

In recent years, a number of investigators have studied the phase equilibria of simple fluids in pores by the application of density functional theory. Semina] studies were carried out by Evans and his co-workers (1985,1986). Their approach was considered to be the simplest realistic model for an inhomogeneous three-dimensional fluid . The starting point was a model intrinsic Helmholtz free energy functional F(p), with a mean-field approximation for the attractive forces and hard-sphere repulsion. As explained in Section 7.6, the equilibrium density profile of the fluid in a pore was obtained by minimizing the grand potential functional. 

It has proven to be useful to decompose the Helmholtz free energy functional A[p(N)], where p(N) is the N-body particle density, into an ideal contribution from non-interacting particles and an excess contribution,... 

The ideal part of Helmholtz free energy functional can be obtained by integrating the ideal chemical potential in its local form, namely // (r) = feB T" In (p(r) A ), over the one-body density distribution p(r), and it follows... 

Application of DFT as a general methodology to classical systems was introduced by Ebner et al. (1976) in modeling the interfacial properties of a Lennard-Jones (LJ) fluid. The basis of all DFTs is that the Helmholtz free energy of an open system can be expressed as a unique functional of the density distribution of the constituent molecules. The equilibrium density distribution of the molecules is obtained by minimizing the appropriate free energy. 

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. 

The functional Eq. (6.33) suggests that it is most natural to consider the Helmholtz free energy as a functional of the external potential energy field A = A[(Pa(r)]. Our interest will he in density functionals. To pursue this we introduce the Legendre transform (Gallon, 1985), A — fv a(f)Pa(f ) = [Pa(f)]-... 

The condition for equilibrium may be described by any of several thermodynamic functions, such as the minimization of the Gibbs or Helmholtz free energy or the maximization of entropy. If one wishes to use temperature and pressure to characterize a thermodynamic state, one finds that the Gibbs free energy is most easily minimized, inasmuch as temperature and pressure are its natural variables. Similarly, the Helmholtz free energy is most easily minimized if the thermodynamic state is characterized by temperature and volume (density) [4]. 

As we saw above, what emerges from our detailed analysis of the vibrational spectrum of a solid can be neatly captured in terms of the vibrational density of states, p(co). The point of this exercise will be seen more clearly shortly as we will observe that the thermodynamic functions such as the Helmholtz free energy can be written as integrals over the various allowed frequencies, appropriately weighted by the vibrational density of states. In chap. 3 it was noted that upon consideration of a single oscillator of natural frequency co, the associated Helmholtz free energy is... 

In this expression, n(r, single-molecule density and is a function of both position, r, and orientation, co. F[n] is the intrinsic Helmholtz free energy, /i is the chemical potential and (r, a) is an external field. It can be shown (Evans ) that the equilibrium molecular density p(r,co) represents a global minimum in Q[n] with respect to functional variations in the trial function (r, tu) at fixed < (r, m) and /t, a necessary condition for which is that... 

Here D (r) describes the influence of the intramolecular correlations, and cj, (r) gives the effect of intermolecular interactions upon the site density. Even in the absence of the intermolecular effects, the intramolecular correlations make a nontrivial contribution to the intrinsic Helmholtz free energy of the ideal gas. Chandler et al. 76.i77 (jj uss two routes to the calculation of DJr). The first approach involves a second-order functional... 

Tarazona and Navascues have proposed a perturbation theory based upon the division of the pair potential given in Eq. (3.5.1). In addition, they make a further division of the reference potential into attractive and repulsive contributions in the manner of the WCA theory. The resulting perturbation theory for the interfacial properties of the reference system is constructed through adaptation of a method developed by Toxvaerd in his extension of the BH perturbation theory to the vapor-liquid interface. The Tarazona-Navascues theory generates results for the Helmholtz free energy and surface tension in addition to the density profile. Chacon et al. have shown how the perturbation theories based upon Eq. (3.5.1) may be developed by a series of approximations within the context of a general density-functional treatment. 

There are many similarities among the various applications of DFT. One of the main characteristics of this parallelism is the existence of variational principles. Thus for electron densities, the electronic energy is a unique functional of the density for a given external potential. For a fluid of atoms or molecules, the intrinsic Helmholtz free-energy is a unique functional of the density for a given interatomic or intermolecular potential. For a nuclear system, the energy of the nuclei can also be regarded as a functional of the... 

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