The ordering of the terms in the excess free energy, after the mixing entropy terms, reflects the order of terms in the diagrams. First, the transfer matrix contributions are defined as above, with Ξ P ± = ψ i M N P ± P ± ij ψ j and the cluster diagram-derived parameters. The subsequent RPA term is numerically evaluated, so we use a version where the thermodynamically irrelevant term linear in species densities φ i is removed for convenience. The functions