Levy-Lieb constrained-search formulation as a minimization of the correlation functional

The constrained- search formulation of Levy and Lieb, which formally defines the exact Hohenberg - Kohn functional for any N- representable electron density, is here shown to be equivalent to the minimization of the correlation functional with respect to theN- 1 conditional probability density, where N is the number of electrons of the system. The consequences and implications of such a result are here analysed and discussed via a practical example.