  The BCS and/or HFB theories are extended by treating the effect of four
quasi-particle states perturbatively. The approach is tested on the pairing
hamiltonian, showing that it combines the advantage of standard perturbation
theory valid at low pairing strength and of non-perturbative approaches
breaking particle number valid at higher pairing strength. Including the
restoration of particle number, further improves the description of pairing
correlation. In the presented test, the agreement between the exact solution
and the combined perturbative + projection is almost perfect. The proposed
method scales friendly when the number of particles increases and provides a
simple alternative to other more complicated approaches.
