Density functional and classic integral equation theories

2 Density functional and classic integral equation theories 

Our goal in this section is to establish a common density functional language for considering solution thermodynamics, and then to provide some density functional perspective on the chemical potentials that are the principal actors in this book. Those results are the foremost goal of this section. But we then use those results to follow Percus s development (Percus, 1964) of classic integral equations for the structure of hquids, and to make further observations that provide more perspective on these issues. 

A rule that assigns to each function in a suitable set a number in another set is 2l functional (Mathews and Walker, 1964 Stakgold, 1979). According to some practices, a basic view of the theory of solutions is to find, study, and characterize the functionals which yield the thermodynamic properties when the intermolecular 

The functional derivative relations we consider here are defined as follows (Hansen and McDonald, 1976, see Section 4.2 Rowlinson and Widom, 1982, see Section 4.5). Consider a functional A of functions /(x) this will be denoted by A[f x)]. Then consider a small change f f + 8f. An evaluation of the change 

A small handful of functional derivatives underlie the discussions that will follow. Our initial step will be to consider the Helmholtz free energy A, introduced in Eq. (2.15), p. 26, and how it depends on an external potential energy field of the form anticipated by Eq. (6.31). In the absence of such an external field, the Helmholtz free energy is a function of ( , E, 7), A = A n,V,T). Thus we now generalize our questions to consider A[ p r)] = A n, V, T)[ (r)]. Our evaluation will be based upon the classical expression 

---