2 FIG. 1: Effective McMillan-Mayer short-range pair potentials extracted from explicit solvent simulations using the HNC closure. (a) Cation anion, (b) cation cation, (c) anion anion, (d) cation anion RDF obtained from explicit solvent MD and implicit solvent MC simulations. FIG. 2: (Color online) (a) Osmotic coefficient Φ in the McMillan-Mayer frame of reference. (diamond) MC simula- tions, (dot dashed) MSA2, (dot) Debye H¨ uckel Limiting law (DHLL), (cross) experiments (Ref. [18] with the McMillan- Mayer to Lewis Randall conversion). (b) Minimization diam- eters. (dot dashed) MSA2 and (diamond) MSA-fit. 0.9 1 1.1 1.2 1.3 MC MSA2 DHLL Exp 0 0.5 1 1.5 c 1/2 (mol.L$^{-1}$) 1/2 3 4 5 (Å) $_{1}$(MSA-fit) $_{2}$(MSA-fit) $_{1}$(MSA2) $_{2}$(MSA2) (a) (b) r (Å) 0 1 2 3 V$_{12}$ $^{SR}$(r) 2 4 6 8 g$_{12}$$^{MC}$(r) g$_{12}$$^{MD}$(r) 4 6 8 0 1 2 3 V$_{11}$ $^{SR}$(r) 4 6 8 0 1 2 3 V$_{22}$ $^{SR}$(r) (a) (b) (d) (c) pute all ion thermodynamic properties through implicit solvent MC simulations. The second stage of our coarse-graining procedure con- sists in applying LPT, in order to deduce the best ana- lytical model of electrolyte solutions which reproduces this molecular description. The principle of LPT is to describe the properties of a given system in terms of those of a well known reference system, with the differ- ence between them treated as a perturbation in the ref- erence potential. Assuming pairwise additive potentials, V$_{ij}$ = V (0) ij + ΔV$_{ij}$ , a first-order truncated expression for the free energy density of the system βf$_{v}$ is obtained, which depends only on the free-energy density f (0) v and RDF g (0) of the reference fluid, with β = ( k$_{B}$T ) − 1 and ρ$_{i}$ the concentration of species i . The Gibbs-Bogoliubov inequality [15] ensures that the right-hand side of Eq. (1) is actually a strict upper bound. Once a reference system has been chosen, the expression on the right-hand side of Eq. (1) must be minimized with respect to the parameters defining the reference. This procedure yields the best first-order approximation to the free energy of the system under consideration. For a system of charged particles in solution, the nat- ural reference is the PM, defined in terms of the charge and diameter ( σ$_{i}$ ) of each species. In this case, the per- turbing potentials are just the short-range effective po- tentials computed above (Δ V$_{ij}$ = V SR ij ). We use the MSA [3] solution to the PM, since it provides analyti- cal expressions for both the free energy and the RDF. The perturbation term is evaluated using an exponential approximation to the RDF obtained within the MSA, g ( r ) = exp [ g$_{MSA}$ ( r ) − 1], which removes any unphysical negative regions and improves the comparison with HNC calculations. We first used LPT for a two-component system (Na + and Cl − free ions) within the MSA (model MSA2), for concentrations ranging from 0.1 to 2 . $_{0 mol l}$− $^{1}$. The mini- mization leads to almost constant diameters on the whole range of concentration: σ$_{1}$ = 3 . 67 ˚ A and σ$_{2}$ = 4 . 78 ˚ A. As shown in Fig. 2, these parameters yield osmotic co- efficients close to MC calculations only at very low con- centration, i.e., c ≤ 0 . $_{1 mol l}$− 1 (experimental values are given for indicative purposes only, since a perfect model will exactly match the MC results). For molar solutions, the LPT results differ considerably from MC calculations. This discrepancy can easily be understood by comparing the diameters found within the MSA2 calculation with the effective potentials given in Fig. 1. The anion/cation contact distance obtained within the MSA2 calculation is 4 . 2 ˚ A, which is in the region of the second minimum of the effective potential and corresponds to the situation where there is a single layer of water molecules between the ions. The first minimum of the potential, which cor- responds to the contact ion pair (CIP) is thus completely ignored by the MSA2 calculation. If the MSA diameters are directly fitted to reproduce the MC osmotic pres- sure, much smaller values are obtained. These MSA-fit hydrated diameters, which are compared to the MSA2 diameters in the bottom part of Fig. 2, are averages of the CIP and the solvent-separated ion pair. To overcome this difficulty, we have explicitly intro- duced the CIP in our model (species 3). Straightforward calculations, based on a characteristic-function formal- ism, allow us to define an equivalent model in which the free ions and the CIP are explicitly taken into ac- count [19, 20]. We apply this formalism by defining a pair as an anion and a cation at a distance less than 4 ˚ A, which corresponds to the position of the effective potential maximum. The interaction between free, like charges in this new system remains unchanged, and the cation-anion interactions are easily approximated by ex- βf$_{v}$ ≲ βf (0) v + 1 2 β ∑ i,j ρ$_{i}$ρ$_{j}$ ∫ d r g (0) ij ( r ) ΔV$_{ij}$ ( r ) (1)