It should be noted that this implies that Slater determinants built from the single-particle are eigenfunctions of the non-interacting many-body Hamiltonian $$ \begin{equation} \left( \sum_{i=1}^N -\frac{1}{2}\nabla_i^2 + v(r_i) \right) \vert\Phi_p\rangle = E_p \vert\Phi_p\rangle, \tag{4} \end{equation} $$ where \( \vert\Phi_p\rangle = \vert\psi_{p_1\rangle\cdots \psi_{p_N}} \) and $$ \begin{equation} E_p = \sum_{i=1}^N \epsilon_{p_i}. \tag{5} \end{equation} $$