W,C,U . The most natural representations of the terms within a cluster expansion are obtained through the use of cluster diagrams (when describing mixtures, we refer to them instead as "mixture diagrams"). We will often present the relevant diagrams as a visual aid because mixture diagrams are the simplest way to depict the relevant couplings within a free energy calculation (which may include alchemical states), and the derivation of equations often follow easily by inspection of diagrams. We now briefly review the use of mixture diagrams as they relate to our calculations.