In this limit, the obvious expansion parameter is .
The unperturbed ground-state energy,
,
of an even-even
system with particle number N=2n (n stands for the number of
pairs) is that of Eq. (2), i.e.,
In the case of an odd system with N=2n+1, the analogous expressions can be obtained by the following simple modifications. First, in the zero order, the (n+1)-th level is occupied (blocked) by one particle, and hence the single-particle energy e_{n+1} should be added to . Moreover, since the pairing Hamiltonian does not couple orbitals occupied by one nucleon (the blocking effect), the orbital containing the odd particle must be excluded from the sum in Eq. (20), and the number of pairs in Eqs. (19) and (20) must become n=(N-1)/2.
Adding together the zero-, first-, and second-order contributions to
the binding energy, one obtains
the corresponding expressions for
:
Finally, for the single-particle energy-splitting filters (7)
and (8),
one obtains: