Figure 1.

Analysis of unrestrained (gas-phase) simplified ion-ligating models. (a) Helmholtz free energy of selectivity versus the number of (NMA, formamide, or water) molecules, N_I, in the cluster – calculated as $\mathrm{\Delta }\mathrm{\Delta }F=\mathrm{\Delta }{F}_{\text{model}}^{\text{K}\to \text{Na}}-\mathrm{\Delta }{F}_{\text{bulk}}^{\text{K}\to \text{Na}}$ where $\mathrm{\Delta }{F}_{\text{model}}^{\text{K}\to \text{Na}}$ and $\mathrm{\Delta }{F}_{\text{bulk}}^{\text{K}\to \text{Na}}$ are free energies to alchemically transform K⁺→Na⁺ in the cluster and in bulk water, respectively. ΔΔF >0 and ΔΔF < 0 indicate K⁺ and Na⁺ selectivity, respectively. For comparison we show experimental values for water molecules¹⁹,²⁰ (blue dashed/dotted lines). Error bars were obtained by block averaging (see SI). (b) Average coordination number, N_C, versus the number of molecules, N_I, included in the cluster. Vertical bars represent standard deviation of the sample.