Fermi-hypernetted-chain methods and the ground state of fermion matter

The convergence properties of the Fermi-hypernetted-chain method as originated by Fantoni and Rosati are investigated. Numerical results are reported for liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and two model fermion liquids. It turns out that for not too high densities and not too long-ranged correlation functions the convergence to an upper bound for the ground-state energy is excellent, but that for higher densities and/or long-ranged correlation functions, it is easily possible to underestimate the upper bound if one does not apply certain convergence criteria and associated error estimates.