In molecular mechanical force fields, a monatomic ion is represented as a charged van der Waals particle that interacts with all atoms of the molecular system. The Coulomb potential normally specifies the electrostatic interaction of the charge. The Lennard-Jones (LJ) potential expresses the pair-wise exchange-repulsion and dispersion interactions between atoms i and j with the well depth ε ij and the distance R min,ij at which the potential reaches the minimum. Following the Lorentz-Berthelot

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... ombination rule, the charge and the dispersion parameters, ε and R min , define the force field of a monatomic ion. 1 The force field parameters are optimized to reproduce experimental findings through molecular simulations. Various experimental and theoretical data have been considered; a partial list includes the free energy of hydration, the entropy of hydration, the radial distribution function of water oxygen atoms around the ion, the lattice constant and the lattice energy of ionic crystals, osmotic coefficients, and geometric and energetic data from ab initio quantum mechanical calculations of monohydrates to water clusters of ions. 2-8 Unfortunately, it has been difficult to determine a unique set of force field parameters in most approaches even with multiple target observables. This communication elaborates the parameter correlation problem 9 and demonstrates that the non-unique parameters based on the hydration free energy can equivalently represent the thermodynamics of the a monoatomic cation. The hydration free energies, ΔG hyd , of monovalent cations are calculated by performing thermodynamic perturbation simulations of the ion solution. The system contains the ion at the origin of a water sphere of radius 1.6 nm under a solvent boundary potential. 10 The sphere includes 572 TIP3P 11 water molecules equilibrated at 298.15 K and 1 bar. The calculation is performed in several steps. First, one mole of ideal gas of volume 0.024788 m 3 is confined into the volume of 0.001 m 3 at 298.15 K and 1 bar. The entropic contribution is ΔG 1 = -RT ln(0.001/0.024788) = 7.958 kJ mol -1 , where R is the gas constant and T is the absolute temperature. Second, the van der Waals particle with the distance parameter R min " and ε = 0.0 is inserted at the center of a bulk water system. The hydration free energy ΔG 2 of the uncharged fictitious atom is calculated by thermodynamic perturbation with the soft core potential 12 where ε is increased to the value of the particle in steps of 41.84 J mol −1 . Third, the van der Waals particle of given R min and ε is charged to +1e in 11 steps; 0. 05, 0.15, 0.25, 0.35, 0.45, 0.55, 0.65, 0.75, 0.85, 0.95, and 1.0. The charging steps yield the free energy change ΔG 3 by using the weighted histogram analysis method. 13 Langevin molecular dynamics at constant temperature and pressure is performed for 20 ps per each window with the time step of 1 fs. The last 17.5 ps of each window are used to compute the free energy change. The Langevin friction coefficient is set to 25 ps -1 for TIP3P oxygen atoms. The recommended nonbonding cutoff options are electrostatic switching and van der Waals switching functions with cut-on, cutoff, cut-nonbond distances of 1.8, 2.0 and 2.2 nm, respectively. O-H and H-H distances of water are fixed by applying the SHAKE constraint. 14 The dynamics trajectory is saved every 0.1 ps to calculate the radial distribution functions (RDFs). All calculations were performed with the CHARMM program version c35b1. 15 ΔG hyd of monovalent cations are obtained as a function of a range of LJ parameters: R min , spanning from 40 pm to 480 pm with the interval of 40 pm and ε spanning from 0.08368 kJ mol −1 to 1.00416 kJ mol −1 with the interval of 0.08368 kJ mol −1 . The experimental hydration free energy of the sodium ion 16 −365 kJ mol −1 is interpolated from the hydration free energy values to determine the distance parameter that pairs with the well depth parameter scanned from 83.68 J mol −1 to 1,004.16 J mol −1 with the interval of 83.68 J mol −1 . The cubic spline interpolation procedure yields a series of Table 1 . Calculated hydration free energies of the sodium ion a