The expansion process employed to obtain the packing part of the free energy. The expanding potentials (labelled ɛ on the left axis) are shown for 5 evenly spaced values of γ (that controls the cavity size during the expansion process, see Sec. IV below) from left to right, together with the corresponding cavity/water-oxygen radial distribution functions (rdf's, right axis). The data is for a λ value of 6.6 Å. The γ = 1 WCA potential is the rightmost (black) curve.
The TI paths for growing cavities (a) in the pure solvent (b) around the Na+ and Cl− ions, when λ = 6.6 Å. In (a), the integrand (squares) and the cumulative integral (smooth solid curve) for the packing contribution are presented. In (b), the ion-specific curves for the inner-shell contribution for the two ions are presented. The cumulative integrals show that the growing-in of the cavities is a smooth process for the chosen potential.
Solute/water-oxygen rdfs (left axis) and hydration numbers (right axis) for all ions studied. The data for cations is shown in (a), while the data for anions is shown in (b).
Free energy contributions for the Cl− ion vs. λ. Upward triangles are the packing term, and the line through the data is the SPT fit. Open circles are the inner-shell term, while open blue downward triangles are the long-ranged term computed with the midpoint-rule approximation. Blue squares are the sum of the inner-shell and packing terms, illustrating the cancellation point at 3.48 Å. The horizontal black line at −87.6 kcal/mol is the exact numerical intrinsic free energy, , while the blue x's are the computed totals (using the midpoint-rule approximation for the long-ranged term).
Free energy data for cations, showing the inner-shell and total free energy data. The circles are for the inner-shell terms, and the lines through the data are fit to the larger cavity data (with a consistent slope of roughly −20.3). The horizontal lines are the exact results (intrinsic free energy): −61.8, −72.4, −88.8, and −113.9 kcal/mol for the Cs+, K+, Na+, and Li+ ions, respectively. The x's are the numerical results computed using the midpoint-rule approximation for the long-ranged term. The squares locate the length at which the inner-shell and packing terms cancel.
Free energy data for anions, showing the inner-shell and total free energy data. The circles are for the inner-shell terms, and the lines through the data are fit to the larger cavity data (with a consistent slope of roughly −21.9). The horizontal lines are the exact results (intrinsic free energy): −72.2, −81.0, −87.6, and −117.2 kcal/mol for the I−, Br−, Cl−, and F− ions, respectively. The x's are the numerical results computed using the midpoint-rule approximation for the long-ranged term. The squares locate the length at which the inner-shell and packing terms cancel.
Data for the long-ranged contribution for the Na+ (top) and Cl− (bottom) ions. The open black circles are for the uncoupled sampling fluctuation term, while the open black squares are for the coupled sampling fluctuation term (right axis). The green x's are for the long-ranged term computed with the midpoint rule (left axis). The blue +'s are for the trapezoid-rule approximation (average of two mean-field bounds), and the red stars are for the average of the two Gaussian approximations.
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