Optimal correlation function for Fermi HNC equations

We show how to find variational equations for the Jastrow pair-correlation function which minimizes the energy of a normal Fermi liquid in the hypernetted chain approximation. By a straightforward extension of the method of Lagrange multipliers, a set of coupled, non-linear integrodifferential equations is obtained. The solutions give both the trial function and the corresponding radial distribution function. For bosons, our equations coincide with the results of paired-phonon analysis.