Pair correlation functions average energy

Now making use of the relationship between the average potential energy (W) and the pair correlation function (equation 2.8.8), one obtains... 

The compressibility equation involves the radial distribution function even when the system consists of nonspherical particles. We recall that previously obtained relations between, say, the energy or the pressure, and the pair correlation function were dependent on the type of particle under consideration. The compressibility depends only on the spatial pair correlation function. If nonspherical particles are considered, it is understood that g(R) in (3.109) is the average over all orientations (3.105). In the following, we shall remove the bar over g(R). We shall assume that the angle average has been taken before using the compressibility equation. 

Here u. j(r,T,P) is the short-range potential for ions, and Eq is the dielectric constant of the solvent. The solvent averaged potentials are thus actually free energies that are functions of temperature and pressure. The thermodynamic properties calculated from the pair correlation functions are summarized below. 

As well as the free energy, the average energy itself is readily calculated from /o. To obtain other averages, such as pair correlation functions, obviously requires more elaborate filing of data during the Markov chain, but this can be done if necessary. 

Unfortunately, it is not easy to calculate the solute-solute pair correlation function in the cluster model. Nevertheless, we can predict the results that could have been obtained in the cluster model from the following considerations. Recall that the indirect part of the potential of average force is related to the solvation Gibbs energies of two monomers and dimer by... 

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