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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. 

## APPLICATION TO HYDRATION OF HARD SPHERES 

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. 

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## APPLICATION TO HYDRATION OF INERT GAS ATOMS 

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 

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where X$_{1}$ is the mole fraction solubility in water and n$_{w}$ is the number density of liquid water. 

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 ). 

