Subtraction of the spurious translational mode from the RPA response function

It is well known that within self-consistent random-phase approximation (RPA) on top of Hartree-Fock (HF), the translational symmetry should be restored. Because of approximations at the level of the practical implementation, this restoration may be only partial. As a result, one has spurious contributions in the physical quantities that are extracted from RPA. Although there are several recipes in the literature to overcome this drawback to produce transition densities or strength functions that are free from spurious contamination, there is no formalism associated with the full RPA response function. We present such formalism in this paper. Our goal is to avoid spurious contamination when the response function is used in many-body frameworks like the particle-vibration coupling theory.