Since our Hamiltonian only has one body terms. We will truncate to \( T=T_1 \) where $$ \begin{align} T_1=\sum_{ia}t_i^aA_a^{\dagger}A_i. \tag{10} \end{align} $$ Thus, we define our ansatz as $$ \begin{align} \vert\Psi(\theta)\rangle=\exp\left\{\sum_{ia}t_i^a\left(A_a^{\dagger}A_i-A_aA_i^{\dagger}\right)\right\}\vert\Phi\rangle. \tag{11} \end{align} $$ We define the set of angles \( \theta=\{t_i^a \ | \ i < F, \ a \geq F\} \) where \( F \) is the number of particles below the Fermi level.