Analysis of Hartree-Fock equations and Koopman's theorem

We can thus make our first interpretation of the separation energies in terms of the simplest possible many-body theory. If we also recall that the so-called energy gap for neutrons (or protons) is defined as $$ \Delta S_n= 2BE(N,Z)-BE(N-1,Z)-BE(N+1,Z), $$ for neutrons and the corresponding gap for protons $$ \Delta S_p= 2BE(N,Z)-BE(N,Z-1)-BE(N,Z+1), $$ we can define the neutron and proton energy gaps for \( {}^{16}\mbox{O} \) as $$ \Delta S_{\nu}=\epsilon_{0d^{\nu}_{5/2}}^{\mathrm{HF}}-\epsilon_{0p^{\nu}_{1/2}}^{\mathrm{HF}}, $$ and $$ \Delta S_{\pi}=\epsilon_{0d^{\pi}_{5/2}}^{\mathrm{HF}}-\epsilon_{0p^{\pi}_{1/2}}^{\mathrm{HF}}. $$