An approach for particle-hole correlation functions, based on the so-called
SCRPA, is developed. This leads to a fully self-consistent RPA-like theory
which satisfies the f-sum rule and several other theorems. As a first step, a
simpler self-consistent approach, the renormalized RPA, is solved numerically
in the one-dimensional Hubbard model. The charge and the longitudinal spin
susceptibility, the momentum distribution and several ground state properties
are calculated and compared with the exact results. Especially at half filling,
our approach provides quite promising results and matches the exact behaviour
apart from a general prefactor. The strong coupling limit of our approach can
be described analytically.Comment: 35 pages, 18 Figures, Feynman diagrams as 10 additional eps-files,
revised and enhanced version, accepted in Phys. Rev.