The first series of diagrams, underlined in red, are those involved with the transfer matrix theory. For brevity, we now implicitly include the integrations over the molecular degrees of freedom and the proximity constraints, and we explicitly include only the diagrams for the polycation species with the polyanion species remaining implied. These diagrams include both the polymer-salt diagrams, as well as the series of 'ladder diagrams' that reflect the multi-site pairing between adjacent chains. These are third-order terms, because the corresponding second-and first-order terms are included in the RPA portion of the free energy, which is given by the ring diagrams that are underlined in dark blue. For this work, we will use the Olvera de la Cruz model, since like the TM calculation this incorporates some elements of the finite particle size in a pairwise fashion. Finally, the third contribution, underlined in light blue, is the third and higher-order diagrams associated with the hard core repulsions of all the species. We deliberately neglect the second-order contribution, as it is already included in the RPA and transfer matrix portions of the calculation. We can use the models discussed in the previous section to write a free energy expression for coacervation using all of these diagrams: