text: we have no additional information, we consider only sym- 
  metric dumbbells. Furthermore, since analytic expres- 
  sions for the RDF within BIMSA are not known, we ap- 
  proximate the dumbbell as a hard sphere when comput- 
  ing the perturbation term (this is not necessary for the 
  reference term, since an expression for the free energy 
  is available). Let 
  σc be the diameter of the cation (an- 
  ion) within the dumbbell, the diameter of the hard sphere 
  e 
  σc[21]. 
  representing this dumbbell is taken to be 
  three- 
  e 
  component MSA3 and BIMSA3, we obtain results in 
  much better agreement with the MC simulations, as 
  shown in Fig. 4. The diameters obtained for species 1, 
  2, and 3 are 3.65, 4.79, and 5.76 ˚A for MSA3 and 3.69, 
  4.75 and 6.19 ˚A for BIMSA3. The free ion diameters are 
  similar for MSA2, MSA3, and BIMSA3. The pair diam- 
  eter is smaller when modeled as a hard sphere (MSA3) 
  than when modeled as a dumbbell (BIMSA3). At high 
  1), the MSA3 overestimates 
  concentration (about 1 mol l− 
  the free energy, because the excluded volume repulsion 
  becomes too important for the pairs to be represented as 
  hard spheres. The BIMSA3 model is the closest to the 
  MC simulation results. It is worth noting that even at 
  the lowest concentration considered, the fraction of pairs 
  (shown in the insert of Fig. 4), although less then 5%, 
  has a non-negligible eﬀect on the thermodynamics of the 
  system. 
  
  σ3 = 4√2 
  π 
  the  Using these  two reference 
  systems, 
  e 