Accuracy of BCS-based approximations for pairing in small Fermi systems

We analyze the accuracy of BCS-based approximations for calculating correlation energies and odd-even energy differences in two-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.