  Giant Resonances are, with nuclear rotations, the most evident expression of
collectivity in finite nuclei. These two categories of excitations, however,
are traditionally described within different formal schemes, such that
vibrational and rotational degrees of freedom are separately treated and
coupling effects between those are often neglected. The present work puts
forward an approach aiming at a consitent treatment of vibrations and
rotations. Specifically, this paper is the last in a series of four dedicated
to the investigation of the giant monopole resonance in doubly open-shell
nuclei via the ab initio Projected Generator Coordinate Method (PGCM). The
present focus is on the treatment and impact of angular momentum restoration
within such calculations. The PGCM being based on the use of deformed
mean-field states, the angular-momentum restoration is performed when solving
the secular equation to extract vibrational excitations. In this context, it is
shown that performing the angular momentum restoration only after solving the
secular equation contaminates the monopole response with an unphysical coupling
to the rotational motion, as was also shown recently for (quasi-particle)
random phase approximation calculations based on a deformed reference state.
Eventually, the present work based on the PGCM confirms that an a priori
angular momentum restoration is necessary to handle consistently both
collective motions at the same time. This further pleads in favor of
implementing the full-fledged projected (quasi-particle) random phase
approximation in the future.
