The exchange of the ions is represented by removing the native ion A from the complex, which was initially geometry optimized with an energy of 𝐸 *%+,#"-.$%& 4 , and solvating it with four water molecules, geometry optimized with an energy of 𝐸 1%#2)("3 $%& 4 . The initially "unbound" ion B, which was geometry optimized with an energy of 𝐸 1%#2)("3 $%& 0 , is taken from a solvated configuration and placed into the molecular complex for geometry optimization to energy 𝐸 *%+,#"-.$%& 0 ,. A solvation shell of four water molecules was chosen to provide a consistent solvation coordinating environment for each of the three ions because this is the largest number of waters that can be coordinated to the cation consistently throughout a geometry optimization (Supporting Information Figures and). This differential binding energy, DEtotal, provides a measure of the relative strength of interaction between the complex and each of the ions and hence can be used as an indicator of the complex's selectivity of certain ions relative to other ions.