This means that we can actually write an ansatz for the ground state of the system as a linear combination of terms which contain the ansatz itself $|c\rangle$ with an admixture from an infinity of one-particle-one-hole states. The latter has important consequences when we wish to interpret the Hartree-Fock equations and their stability. We can rewrite the new representation as $$|c'\rangle = |c\rangle+|\delta c\rangle,$$ where $|\delta c\rangle$ can now be interpreted as a small variation. If we approximate this term with contributions from one-particle-one-hole (1p-1h) states only, we arrive at $$|c'\rangle = \left(1+\sum_{ai}\delta C_{ai}a_{a}^{\dagger}a_i\right)|c\rangle.$$ In our derivation of the Hartree-Fock equations we have shown that $$\langle \delta c| \hat{H} | c\rangle =0,$$ which means that we have to satisfy $$\langle c|\sum_{ai}\delta C_{ai}\left\{a_{a}^{\dagger}a_i\right\} \hat{H} | c\rangle =0.$$ With this as a background, we are now ready to study the stability of the Hartree-Fock equations.