Convergence of hypernetted-chain calculations for extended Fermi systems

We investigate the convergence properties of various hypernetted-chain approximations for the calculation of the ground state energy of an infinite Fermi system. Our considerations involve several levels of a previously invented Fermi-hypernetted-chain (FHNC) approximation as well as the Wu-Feenberg (Pandharipande-Bethe) method of incorporating Fermi statistics by a cluster expansion for the antisymmetry corrections. From analytical as well as numerical considerations it is shown that - if a sophisticated many-body method is needed at all - a suitable arrangement of cluster corrections is indispensable. We demonstrate in particular that an apparent fast convergence of partial series may well be simulated by an improper classification.