The hydration number of Na+ in liquid water
  An `ab initio' molecular dynamics simulation of a Na+ ion in aqueous solution is presented and discussed. The calculation treats a Na+ ion and 32 water molecules with periodic boundary conditions on a cubic volume determined by an estimate of zero partial molar volume for this solute in water at normal density and at a temperature of 344+/-24 K. Analysis of the last half of the 12 ps trajectory shows 4.6 water molecules occupying the inner hydration shell of the Na+ ion on average, with 5 being the most probable occupancy. The self-diffusion coefficient observed for the Na+ is 1.0 x 10$^{-5}$ cm$^2$/s. The quasi-chemical theory of solutions provides the framework for two more calculations. First a complementary calculation, based on electronic structure results for ion-water clusters and a dielectric continuum model of outer sphere hydration contributions, predicts an average hydration shell occupancy of 4.0. This underestimate is attributed to the harmonic approximation for the clusters in conjunction with the approximate dielectric continuum model treatment of outer sphere contributions. Finally, a maximum entropy fitting of inner sphere occupancies that leads to insensitive composite free energy approximations suggests a value in the neighborhood of -68 kcal/mol for the hydration free energy of Na+ (aq) under these conditions with no contribution supplied for packing or van der Waals interactions. 
Susan B. Rempe and Lawrence R. Pratt