It is possible to write this partition function in a more concise way via the well-known 'cluster expansion': Here, the circles are 'z-circles' that indicate a contribution from factors z * α (i) for each circle in the diagram, and the connecting gray lines are 'e-bonds' that represent the Boltzmann factors e α,γ (i, j) for each pair of particles. 118 Note that in this four-component model, each circle implies a summation over all possible species α, which in this case includes all of the charged (i.e. non-water) particles. Therefore, we also consider all combinations of e α,γ (i, j) between each pair of factors z * α (i) and z * γ (j). For example, the fourth (i.e. three z-circle) term in this expansion can be written as: