Particle-number conservation within self-consistent random-phase approximation

The self-consistent random-phase approximation (SCRPA) is reexamined within a multilevel-pairing model with double degeneracy. It is shown that the expressions for occupation numbers used in the original version of SCRPA violate the particle number for non-symmetric particle-hole ($ph$) spectra. A renormalization is introduced to restore the particle number, which leads to the expressions of occupation numbers similar to those derived by Hara et al. for the ph case. The results of calculations within the $ph$-symmetric case show that this number-conserving SCRPA yields the energies of the ground state and first excited state of the system with $\Omega+2$ particles relative to the ground state of the system with $\Omega$ particles in close agreement with those obtained within the original SCRPA. However it gives a slightly larger correlation energy.