We wish now to derive the Hartree-Fock equations using our second-quantized formalism and study the stability of the equations. Our ansatz for the ground state of the system is approximated as (this is our representation of a Slater determinant in second quantization) $$ |\Phi_0\rangle = |c\rangle = a^{\dagger}_i a^{\dagger}_j \dots a^{\dagger}_l|0\rangle. $$ We wish to determine \( \hat{u}^{HF} \) so that \( E_0^{HF}= \langle c|\hat{H}| c\rangle \) becomes a local minimum.

In our analysis here we will need Thouless' theorem, which states that an arbitrary Slater determinant \( |c'\rangle \) which is not orthogonal to a determinant \( | c\rangle ={\displaystyle\prod_{i=1}^{n}} a_{\alpha_{i}}^{\dagger}|0\rangle \), can be written as $$ |c'\rangle=exp\left\{\sum_{a>F}\sum_{i\le F}C_{ai}a_{a}^{\dagger}a_{i}\right\}| c\rangle $$