The LR contribution is estimated from the two distributions of solute-solvent ion interaction energies, which are calculated over configurations from uncoupled and coupled simulations. As shown in panel a of Fig. for the solute Na + , the mean uncoupled interaction energy is -125.62 kcal/mol with a fluctuation contribution (Eq. 8) of 28.2 kcal/mol, while the mean coupled interaction energy is -180.81 kcal/mol with a fluctuation contribution of 21.3 kcal/mol. The difference between the two fluctuation terms leads to an error of 3.5 kcal/mol for the LR free energy contribution, indicating that the Gaussian approximation is inappropriate and a larger cavity is required to observe Gaussian behaviour as was previously seen in the calculation of the hydration free energy of the Na + ion in water. As discussed in the theoretical methods section above, however, the long-range contribution is equal to the interaction energy at the intersection point of the two distributions. Thus we estimate the long-ranged contribution for Na + as -156.0 kcal/mol. In panel b of Fig. , the distributions for the Cl -ion exhibit more closely Gaussian behavior since the fluctuation terms are closer: 25.2 kcal/mol for uncoupled simulation with a mean value of 84.05 kcal/mol and 27.6 kcal/mol for coupled simulation with mean value of 21.40 kcal/mol.