  We analyze the accuracy of BCS-based approximations for calculating
correlation energies and odd-even energy differences in 2-component fermionic
systems with a small number of pairs. The analysis is focused on comparing BCS
and projected BCS treatments with the exact solution of the pairing
Hamiltonian, considering parameter ranges appropriate for nuclear pairing
energies. We find that the projected BCS is quite accurate over the entire
range of coupling strengths in spaces of up to about 20 doubly degenerate
orbitals. It is also quite accurate for two cases we considered with a more
realistic Hamiltonian, representing the nuclei around 117Sn and 207Pb. However,
the projected BCS significantly underestimates the energies for much larger
spaces when the pairing is weak.
