An alternative to the above approach would be to explicitly include the non-Coulomb portion of the interaction potential, βu s , as sums over some or all of the remaining non-ring diagrams. Perhaps the most trivial class of these diagrams are the diagrams without any Ψ-bonds, and are the diagrams formed only from f 0 -bonds. This is schematically identical to the diagram expansion for the excess free energy using f -bonds, though neglecting any contribution due to the long-range electrostatics. If these f -bonds are single-site (i.e. each monomer is considered unconnected), and u s is the hard-core potential, then this would lead to the well-known Carnahan-Starling result: 118,123