Projection and ground state correlations made simple

We develop and test efficient approximations to estimate ground state correlations associated with low- and zero-energy modes. The scheme is an extension of the generator coordinate method (GCM) within Gaussian overlap approximation (GOA). We show that the GOA fails in non-Cartesian topologies and presents a topologically correct generalization of the GOA (topGOA). A random-phase-approximation like (RPA-like) correction is derived as the small amplitude limit of topGOA, called topRPA. Using exactly solvable models, the topGOA and topRPA schemes are compared with conventional approaches (GCM-GOA, RPA, Lipkin-Nogami projection) for rotational-vibrational motion and for particle-number projection. The results shows that the new schemes perform very well in all regimes of coupling.