Summarizing and defining a normal-ordered Hamiltonian

$$ \left\{a_p^\dagger, a_q \right\}= \delta_{pq}, p, q \leq \alpha_F $$ $$ \left\{a_p, a_q^\dagger \right\} = \delta_{pq}, p, q > \alpha_F $$ with \( i,j,\ldots \leq \alpha_F, \quad a,b,\ldots > \alpha_F, \quad p,q, \ldots - \textrm{any} \) $$ a_i|\Phi_0\rangle = |\Phi_i\rangle, \hspace{0.5cm} a_a^\dagger|\Phi_0\rangle = |\Phi^a\rangle $$ and $$ a_i^\dagger|\Phi_0\rangle = 0 \hspace{0.5cm} a_a|\Phi_0\rangle = 0 $$