An extension of the modified weighted-density approximation (MWDA) is presented which retains the key features of the original MWDA in that it continues to describe the nonuniform system through the use of low-order correlation functions of the uniform counterpart. However, the approximate free energy functional is now exact up to third order in the functional expansion of the free energy, and therefore requires as input both the second- and third-order direct correlation functions of the uniform liquid, as well as its excess free energy per particle. The theory has been applied previously to the problem of isochoric freezing of the classical one-component plasma, and is here applied to the well-known problem of isobaric freezing of hard spheres. We use two different approaches to describe the less well-known third-order direct correlation function of the uniform liquid. The first approach is representative of a class of models for this function that are derived through three functional density differentiations of an approximate free energy model. The second is a factorization ansatz based on liquid-state diagrammatic expansions. The results are quite sensitive to these choices: The first leads to an improvement over the already satisfactory results of the original MWDA for the hard-sphere system, whereas the second does not lead to freezing at all. These differences are traced to the ways in which the approximations treat long-range and short-range potentials.  

Likos, C. N., & Ashcroft, N. W. (1993). Density-functional theory of nonuniform classical liquids: An extended modified weighted-density approximation. The Journal of Chemical Physics, 99(11), 9090–9102.