Free-energy functional of the Debye-H\"uckel model of simple fluids

The Debye-H\"uckel approximation to the free-energy of a simple fluid is written as a functional of the pair correlation function. This functional can be seen as the Debye-H\"uckel equivalent to the functional derived in the hyper-netted chain framework by Morita and Hiroike, as well as by Lado. It allows one to obtain the Debye-H\"uckel integral equation through a minimization with respect to the pair correlation function, leads to the correct form of the internal energy, and fulfills the virial theorem.