Equivalence of two free-energy models for the inhomogeneous hard-sphere fluid

It is shown that the free-energy model for the inhomogeneous hard-sphere fluid introduced by Rosenfeld [Phys. Rev. Lett. 63, 980 (1989)], which contains both vector and scalar contributions, and the fully scalar expression proposed subsequently by Kierlik and Rosinberg [Phys. Rev. A 42, 3382 (1990)] represent two equivalent forms of the same functional. This unexpected result was partly obscured by a sign error in the original derivation of the vector part.