Conditional solvation and the pair correlation function

In this section, we examine in some more detail the relation between the conditional and the unconditional solvation Helmholtz energies. Consider a solvent l at any given temperature, volume, and composition. Let s be a simple spherical molecule. The solvation Helmholtz energy of s is given by 

Note that we have explicitly introduced the location of Ri at which we have placed the solvation s. This is in general not necessary since all points in the solvent are equivalent. In this section, however, we shall produce inhomogeneity in the solvent by introducing two particles at Ri and R2, therefore, the recording of their locations is important. In this section, we assume that we have a dilute solution of s in a solvent. Theoretically, we can think of having just one s in a pure solvent. The symbol ( )0 stands for an average over all the configurations of the solvent molecules in the T, V, N ensemble, i.e., 

Suppose we have one solute at Ri and we introduce a second solute at a different location R2. The corresponding work is obtained by taking the ratio of the two partition functions 

R2) is the direct interaction between the two solutes at Rx, R2, and the symbol ( )Ri stands for a conditional average, i.e., an average over all configurations of the solvent molecules, given one solute at Rx. The corresponding conditional density is 

the correlation function g(Ri, R2) connects the solvation Helmholtz energy on the first site and on the second site. Equation (7.140) can also be rewritten as 

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