text: 5 
  ) 
  Å 
  4 
  ( 
  σ 
  3 
  
  0  8 
  r (Å) 
  
  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. 
  
  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, 
  Vij = V (0) 
  ij + ∆Vij , a ﬁrst-order truncated expression for 
  the free energy density of the system βfv is obtained, 
  
  1 
  2 
  Xi,j 