text: which depends only on the free-energy density f (0) 
  and 
  v 
  RDF g(0) of the reference ﬂuid, with β = (kBT )− 
  1 and 
  ρi 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 (σi) of each species. In this case, the per- 
  turbing potentials are just the short-range eﬀective po- 
  tentials computed above (∆Vij = 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 [gMSA(r) − 1], which removes any unphysical 
  negative regions and improves the comparison with HNC 
  calculations. 