Restricted variation after projection and the Lipkin-Nogami methods

We analyze the ability of a restricted variation after projection method to achieve the full variation after projection solution in a pairing Hamiltonian. The study of the projected potential energy surfaces defined along the fluctuations of the particle number (ΔN)2 and (ΔN)4 allows the enlargement of the variational space to take into account more correlations within the projection after the variation framework. The results show the equivalence of the full variational projected solution to a restricted projected solution with an educated choice of the restricted variational space. Finally, we show that the Lipkin-Nogami approach can be derived, under certain conditions, in an variational context and that the projected Lipkin-Nogami approach represents an approximation to a restricted variation after projection method with the (ΔN)2 as degree of freedom