The approximate equalities shown along each transformation follow from neglecting the a-b interaction on the free energies. In coupling schemes for which we cannot neglect this kind of effect, the use of partially-connected diagrams (like in the upper-left and lower-right corners) necessitates a generalized cluster expansion, the details of which are out of the scope of the current study. We focus on the pathway for turning on V a,V followed by turning on V b,V , while the other pathway is analogous by symmetry. The free energy change for the first leg of the transformation predominantly includes the effect of coupling a to V :