Vapor density 0 . 023 kg/m 3 Water density 997 . 1 kg/m 3 Bulk modulus 2 . 187 GP a ∂B/∂P 5 . 83 a 6 . 630 × 10 9 b − 1 . 277 × 10 8 c 9 . 200 × 10 5 d − 2 . 602 × 10 3 e 8 . 906 × 10 − 4 f − 1 . 415 × 10 − 2 g 1 . 077 × 10 − 10 TABLE I: Properties of uniform bulk water at Standard Am- bient Temperature and Pressure (SATP: T=25 ◦ C, P=100.00 kPa) 2 nal potential V ( ⃗ r ) can be found by minimizing, over all possible thermally averaged molecular densities ρ ( ⃗ r ), the sum of a universal functional and the interaction of that density with the potential, with the density profile ρ ( ⃗ r ) which minimizes the free- energy giving the exact average density at thermal equi- librium. F [ ρ ( ⃗ r )] is said to be a universal functional of density ρ ( ⃗ r ) in the sense that it does not depend what- soever on the external potential V ( ⃗ r ) but only upon the nature of the solvent itself. This yields the significant advantage that once a good approximation is found for the universal functional F [ ρ ( ⃗ r )], it can be used to study the interaction with any possible external potential V ( ⃗ r ), such as that representing one solute or another. and χ ( ⃗ r ) is an interaction kernel which integrates to zero. The first term in Eq. (4), which is known exactly, ensures that the short length-scale properties (to which F$_{ex}$ does not couple) are treated properly. The second term in Eq. (4) ensures that a large-scale phase separation be- tween vapor and liquid is treated correctly and, often, can capture exactly the experimental two-particle corre- lation functions in the bulk liquid[7] for an appropriate convolution kernel in Eq. (5). However, we find that, for the experimentally observed two-particle correlation function in liquid water, the equation for the kernel has no real solutions. The solutions have small imaginary parts and thus cannot be used directly. The real parts of the solution, however, closely resemble a Gaussian. Thus, we employ a Gaussian in Eq. (5) and add the last term in Eq. (4) to ensure that the experimental two-particle correlation function in the bulk liquid is reproduced. Even though F [ ρ ( ⃗ r )] is exact in principle, its analytical form is not known and approximations must be devel- oped. Generally, one begins by separating the functional as where F$_{id}$ [ ρ ( ⃗ r )] is the free energy of ideal gas whose an- alytic form is known exactly , and where F$_{ex}$ [ ρ ( ⃗ r )] is defined as the excess free energy beyond that of the ideal gas and contains all the other terms responsible for correlations among the molecules. Our density-functional theory is inspired by a class of weighted-density functionals[7]. Our method, however, is computationally much simpler in that it does not require computationally demanding self-consistent calculations of the weighted density. On the other hand, our form does allow us to incorporate much of the same physics as [7] and thus find a form which is competitive compu- tationally but more accurate than other, more simplified functionals[10]. To construct the functions and parameters needed in Eq. (4), we begin by noting that, for the uniform liquid, the last term gives zero ( χ has zero integral) and ¯ ρ ( ⃗ r ) = ρ ( ⃗ r ), so that f$_{ex}$ ( ρ ) can be constructed to reproduce the properties of the uniform liquid exactly. In practice, we parameterized f$_{ex}$ ( ρ ) as a sixth-order polynomial with parameters chosen to reproduce, for bulk water at standard ambient temperature and pressure (SATP), the density and bulk modulus of both the liquid and vapor (four parameters), the derivative of the bulk modulus of the liquid with respect to pressure ∂B/∂P , and phase coexistence of the liquid and vapor (equality of grand potential). Table I gives both the fit data and the re- sulting parameters, and Figure 1 presents the final func- tion F ( ρ ). Note that in this work we employ mostly atomic units (a.u.) so that Planck’s constant, the elec- tron mass and the fundamental charge all have numerical value unity (¯ h = m$_{e}$ = e = 1), implying that energies are measured in units of 1 hartree=27.21 eV and distances in units of 1 bohr=0.5291 ˚ A. To construct χ ( ⃗ r ), we work in the Fourier represen- tation. As noted above, χ ( ⃗ r ) integrates to zero so that where F$_{id}$ [ ρ ( ⃗ r )] is given by Eq. (3) and f$_{ex}$ (¯ ρ ( ⃗ r )) is a local function of ¯ ρ ( ⃗ r ), a Gaussian smoothed density, The form we choose to consider is CONSTRUCTION OF CLASSICAL DENSITY-FUNCTIONAL THEORY FOR WATER F [ ρ ( ⃗ r )] = F$_{id}$ [ ρ ( ⃗ r )] + ∫ f$_{ex}$ (¯ ρ ( ⃗ r )) d⃗ r (4) + ∫ ∫ ρ ( ⃗ r ) χ ( ⃗ r − ⃗ r $^{′}$) ρ ( ⃗ r $^{′}$) d⃗ rd⃗ r ′ ¯ ρ ( ⃗ r ) ≡ (2 λ $^{2}$π ) − 3 2 ∫ ρ ( ⃗ r $^{′}$) e −| ⃗ r − ⃗ $_{r}$′ | 2 2 λ 2 d⃗ r $^{′}$, (5) f$_{ex}$ ( ρ ) = aρ 6 + bρ 5 + cρ 4 + dρ 3 + eρ 2 + f ρ + g. (6) Ω$_{0}$ = min ρ ( ⃗ r ) { F [ ρ ( ⃗ r )] + ∫ ( V ( ⃗ r ) − µ ) ρ ( ⃗ r ) d⃗ r } , (1) F$_{id}$ [ ρ ( ⃗ r )] = kT ∫ ρ ( ⃗ r ) ln( ρ ( ⃗ r )) d⃗ r, (3) F [ ρ ( ⃗ r )] = F$_{id}$ [ ρ ( ⃗ r )] + F$_{ex}$ [ ρ ( ⃗ r )] , (2)