  Recently, it has been shown, that the pair density of the homogeneous
electron gas can be parametrized in terms of 2-body wave functions (geminals),
which are scattering solutions of an effective 2-body Schr\"odinger equation.
For the corresponding scattering phase shifts, new sum rules are reported in
this paper. These sum rules describe not only the normalization of the pair
density (similar to the Friedel sum rule of solid state theory), but also the
contraction of the 2-body reduced density matrix. This allows one to calculate
also the momentum distribution, provided that the geminals are known from an
appropriate screening of the Coulomb repulsion. An analysis is presented
leading from the definitions and (contraction and spectral) properties of
reduced density matrices to the Kimball-Overhauser approach and its
generalizations. Thereby cumulants are used. Their size-extensivity is related
to the thermodynamic limit.
