Here, the Coulomb contribution is βu c,α,γ (r i,α , r j,γ ) = λ B z α z γ /|r i,α -r j,γ | and the hard sphere potential is βu hs (r i , r j ) = 0 if |r i -r j | ≥ σ and otherwise βu hs (r i , r j ) = ∞. This allows us to replace a given f -bond with one or zero f 0 -bonds, and any number of interaction bonds (which we will call Ψ-bonds by writing Ψ α,γ (r i , r j ) = -βu c,α,γ ); however, at least one bond must be present in a given diagram. For the pairwise diagram, this means that the diagrams are now: