 0  10  20  30  40  50  60  70  0  1  2  3  4  5  6  7  8  9 Surface tension (mJ/m 2 ) R (Angstrom) 3 FIG. 2: Surface tension of a hard sphere in water at SATP. Solid line denotes the results of our classical density- functional theory calculations. Molecular simulation results for a cavity in SPC/E water are indicated by diamonds [22]. Dashed line corresponds to macroscopic surface tension of wa- ter. FIG. 1: Free energy of water as a function of its density at SATP ˆ χ ( ⃗ k = ⃗ 0) = 0. For ⃗ k = ⃗ 0, we fit to the experimen- tal pair distribution function g ( r ) of bulk uniform wa- ter at SATP[19]. Finally, the Gaussian width parameter λ = 0 . 4415 a.u. has been fit to the surface tension of bulk water 70 mN/m or 4 . 5 × 10 − 5 a.u. We emphasize that in constructing this functional we used only the macro- scopic properties of water. The ultimate motivation for inhomogeneous continu- ous theories of bulk water is to understand solvation of real solutes, not artificial hard boundaries. Within the density-functional framework of Eq. (1), one can simply and rigorously incorporate the effects of any external po- tential V ( ⃗ r ) acting on the liquid, making treatment of real solutes simple in principle, provided a static poten- tial V ( ⃗ r ) accurately describes the interaction of water with the solute. For our next test, we treat a simple but challenging problem for which there is experimental data, the solvation of inert gas atoms. The experimental solubilities of the inert gasses in wa- ter give the corresponding solvation free energies directly. In the dilute limit, the free energy of solvation ΔΩ relates to the mole fraction solubility through the relation To test the accuracy of this approximation we use it to calculate the free energy of a spherical cavity in water, a standard test case used in the literature [10, 13, 20, 21] to explore density variations in water over all possible length scales. The interaction potential with a hard sphere is over-idealized because it depends only on the distance of the water molecule to the hard sphere and does not depend on the orientation. For instance, for a real water molecule it is unclear even what point to take to represent the location of the molecule. The position of the oxygen nucleus is most often taken. Figure 2 compares our results for the surface ten- sion (free energy change per unit area) of a spherical cavity with results of molecular simulations for SPC/E water[22]. The figure verifies that the surface tension ap- proaches in the macroscopic value in the limit of large radii and show the, for smaller radii, the surface ten- sion has a strong dependence on radius. Our results are in agreement with Lum-Chandler-Weeks[13] theory. The good agreement with explicit molecular dynamics simu- lations suggests that this model gives good quantitative description of the hydrophobic effect, a central problem of theoretical chemistry. The experimental data, which Table II summarizes, show an interesting trend. Although the solvation en- ergies of hard spheres increase with size, the solvation energies of inert gas atoms tend to decrease . (Note that for helium, the worst possible case, we estimate quan- tum zero-point effects to be quite small, on the order of 0.005 eV, so that this must be purely an effect of the interactions.) Thus, a simple cavity model for solvation of inert gas atoms is not a good approximation and one must use a more realistic interaction potential V ( ⃗ r ). where X$_{1}$ is the mole fraction solubility in water and n$_{w}$ is the number density of liquid water. ΔΩ = k$_{B}$T log ( n$_{gas}$ X$_{1}$n$_{w}$ ) = k$_{B}$T log ( 1atm k$_{B}$T X$_{1}$n$_{w}$ ) (7) APPLICATION TO HYDRATION OF INERT GAS ATOMS APPLICATION TO HYDRATION OF HARD SPHERES