  The solvation of hydrophobic solutes in water is special because liquid and
gas are almost at coexistence. In the common hypernetted chain approximation to
integral equations, or equivalently in the homogenous reference fluid of
molecular density functional theory, coexistence is not taken into account.
Hydration structures and energies of nanometer-scale hydrophobic solutes are
thus incorrect. In this article, we propose a bridge functional that corrects
this thermodynamic inconsistency by introducing a metastable gas phase for the
homogeneous solvent. We show how this can be done by a third order expansion of
the functional around the bulk liquid density that imposes the right pressure
and the correct second order derivatives. Although this theory is not limited
to water, we apply it to study hydrophobic solvation in water at room
temperature and pressure and compare the results to all-atom simulations. With
this correction, molecular density functional theory gives, at a modest
computational cost, quantitative hydration free energies and structures of
small molecular solutes like n-alkanes, and of hard sphere solutes whose radii
range from angstroms to nanometers. The macroscopic liquid-gas surface tension
predicted by the theory is comparable to experiments. This theory gives an
alternative to the empirical hard sphere bridge correction used so far by
several authors.
