Approximate particle number projection for finite range density dependent forces.

The Lipkin-Nogami method is generalized to deal with finite range density dependent forces. New expressions are derived and realistic calculations with the Gogny force are performed for the nuclei 164 Er and 168 Er. The sharp phase transition predicted by the mean field approximation is washed out by the Lipkin-Nogami approach. A much better agreement with the experimental data is reached with the new approach than with the Hartree-Fock-Bogoliubov one, specially at high spins.