## 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 $$\Delta S_{\pi}=\epsilon_{0d^{\pi}_{5/2}}^{\mathrm{HF}}-\epsilon_{0p^{\pi}_{1/2}}^{\mathrm{HF}}.$$