Koopman's theorem

If we assume that the spectrum does not change when we excite an electron above the ground-state, we can prove that the energy required to add the electron to an excited state, is precisely $\epsilon _k$, the eigenvalues of $H_{HF}$. In order to do that we need to calculate $E^k_{N+1}-E_N$, where we have added an electron in the orbital $k$. The energy can be easily computed to yield:

$$E^k_{N+1}-E_N = \langle \phi_k \vert h\vert\phi_k \rangle + \langle \phi_k\vert 2J - K\vert \phi_k \rangle = \epsilon_k.$$ (107)

If one sums over all occupied states, the two-body term will become twice larger than 107. Thus, the total energy is also given by:

$$E=\sum_k \left[\epsilon_k - \frac{1}{2}\langle k\vert 2J - K\vert k \rangle\right],$$ (108)

which is obviously not the same as the sum over all the eigenvalues.