  This paper deals with the theoretical foundation of effective two-body forces
for the Generator Coordinate Method (GCM) and the projected mean-field method.
The first aim of this paper is to reduce into various local-densities the
in-medium content of a generalized $G$ matrix removing the hard core problem in
this extended context. Then, we consider the possible renormalization of
multi-body forces through a density-dependent two-body interaction in the
context of configuration mixing calculations. A density dependence of the form
$\rho^{\sigma}$, as used in Skyrme and Gogny forces, is successfully
interpreted as doing so when the mixed density is used. Finally, we propose a
simple extension of the Skyrme force dedicated to the calculation of matrix
elements between non-orthogonal product states, which are needed to evaluate
the correlated energy.
