The variational condition for deriving the Hartree-Fock equations guarantees only that the expectation value \( \langle c | \hat{H} | c \rangle \) has an extreme value, not necessarily a minimum. To figure out whether the extreme value we have found is a minimum, we can use second quantization to analyze our results and find a criterion for the above expectation value to a local minimum. We will use Thouless' theorem and show that $$ \frac{\langle c' |\hat{H} | c'\rangle}{\langle c' |c'\rangle} \ge \langle c |\hat{H} | c\rangle= E_0, $$ with $$ {|c'\rangle} = {|c\rangle + |\delta c\rangle}. $$ Using Thouless' theorem we can write out \( {|c'\rangle} \) as $$ \begin{align} {|c'\rangle}&=\exp\left\{\sum_{a > F}\sum_{i \le F}\delta C_{ai}a_{a}^{\dagger}a_{i}\right\}| c\rangle \tag{89}\\ &=\left\{1+\sum_{a > F}\sum_{i \le F}\delta C_{ai}a_{a}^{\dagger} a_{i}+\frac{1}{2!}\sum_{ab > F}\sum_{ij \le F}\delta C_{ai}\delta C_{bj}a_{a}^{\dagger}a_{i}a_{b}^{\dagger}a_{j}+\dots\right\} \tag{90} \end{align} $$ where the amplitudes \( \delta C \) are small.