  Different steps leading to the new functional for pairing based on natural
orbitals and occupancies proposed in ref. [D. Lacroix and G. Hupin,
arXiv:1003.2860] are carefully analyzed. Properties of quasi-particle states
projected onto good particle number are first reviewed. These properties are
used (i) to prove the existence of such a functional (ii) to provide an
explicit functional through a 1/N expansion starting from the BCS approach
(iii) to give a compact form of the functional summing up all orders in the
expansion. The functional is benchmarked in the case of the picked fence
pairing Hamiltonian where even and odd systems, using blocking technique are
studied, at various particle number and coupling strength, with uniform and
random single-particle level spacing. In all cases, a very good agreement is
found with a deviation inferior to 1% compared to the exact energy.
