## Analysis of Hartree-Fock equations and Koopman's theorem

We have $$E[\Phi^{\mathrm{HF}}(N)] = \sum_{i=1}^H \langle i | \hat{h}_0 | i \rangle + \frac{1}{2}\sum_{ij=1}^N\langle ij|\hat{v}|ij\rangle_{AS},$$ where $$\Phi^{\mathrm{HF}}(N)$$ is the new Slater determinant defined by the new basis of Eq. (13) for $$N$$ electrons (same $$Z$$). If we assume that the single-particle wave functions in the new basis do not change when we remove one electron or add one electron, we can then define the corresponding energy for the $$N-1$$ systems as $$E[\Phi^{\mathrm{HF}}(N-1)] = \sum_{i=1; i\ne k}^N \langle i | \hat{h}_0 | i \rangle + \frac{1}{2}\sum_{ij=1;i,j\ne k}^N\langle ij|\hat{v}|ij\rangle_{AS},$$ where we have removed a single-particle state $$k\le F$$, that is a state below the Fermi level.