Here, the summation is over τ MC steps indicated by the index i. As defined before, f α,γ (r) = e -βuα,γ (r) -1. We choose a form of the potential potential, βu P +,-(r) = βu HS (r) -λ B /r, that includes a hard-sphere part βu HS (r) = 0 for r > 2a and βu HS (r) = ∞ for r ≤ 2a, and an unscreened Coulomb part with a magnitude of βu P +,-(2a) = λ B /(2a) at contact. This calculation depends on several choices, including the simulated chain structure via the values of κ, κ θ , and r eq and the choice of θ s,± . For this case, θ s,± = 1 only if s is the monomer closest to the ion. We also apply a cutoff r c , consistent with the criteria for pairing used in our prior work, 102,103 such that θ s,± = 1 only if r s,± < r c . An analogous numerical calculation is performed for the other quantities (B 0,P ± , F ± , and E ± ) using a similar procedure, with the only difference being that for θ s,s we require that monomer s is the closest to s and that the monomer s is closest to s (i.e. they are the closest pair), in addition to that distance being < r c .