text: The excess free-energy density of the original system 
  f ex 
  v plus a 
  βf ex 
  v 
  correction term 
  is that of the three component mixture β 
  
  e 
  
  This procedure also provides an accurate description of 
  the structure over the whole range of concentrations. A 
  development similar to the one that leads to Eq. (2) de- 
  rives the average unpaired RDF from the corresponding 
  paired quantities: 
  
  βf ex 
  
  f ex 
  v − 
  e 
  
  v = β  ρ3 ln K0,  (4) 
  
  e 
  
  which is due to the change in standard chemical potential 
  between the two component and three component mod- 
  els. It should be noted that the fraction of pairs is now an 
  additional parameter in the minimization scheme, which 
  serves to ensure chemical equilibrium. Within this rep- 
  resentation, the pair can be modeled as a hard sphere 
  (MSA3) or as a dumbbell-like CIP (BIMSA3) [4]. Since 
  
  ρiρjgij(k) = 
  +  (5) 
  
  + 
  
  e  e  e 