  The hydration free energies of ions exhibit an approximately quadratic
dependence on the ionic charge, as predicted by the Born model. We analyze this
behavior using second-order perturbation theory. This provides effective
methods to calculating free energies from equilibrium computer simulations. The
average and the fluctuation of the electrostatic potential at charge sites
appear as the first coefficients in a Taylor expansion of the free energy of
charging. Combining the data from different charge states allows calculation of
free-energy profiles as a function of the ionic charge. The first two Taylor
coefficients of the free-energy profiles can be computed accurately from equi-
librium simulations; but they are affected by a strong system-size dependence.
We apply corrections for these finite-size effects by using Ewald lattice sum-
mation and adding the self-interactions consistently. Results are presented for
a model ion with methane-like Lennard-Jones parameters in SPC water. We find
two very closely quadratic regimes with different parameters for positive and
negative ions. We also studied the hydration free energy of potassium, calcium,
fluoride, chloride, and bromide ions. We find negative ions to be solvated more
strongly compared to positive ions of equal size, in agreement with experimen-
tal data. We ascribe this preference of negative ions to their strong interac-
tions with water hydrogens, which can penetrate the ionic van der Waals shell
without direct energetic penalty in the models used. We also find a positive
electrostatic potential at the center of uncharged Lennard-Jones particles in
water, which favors negative ions. Regarding finite-system-size effects, we
show that even using only 16 water molecules it is possible to calculate accu-
rately the hydration free energy of sodium if self-interactions are considered.
