to the genuine one-body part $H_{\text{int}}^{(1)}$, one obtains a s.p. Hamiltonian $H_{\text{int}} ^{\text{s.p.}}$ whose matrix elements in the natural-orbital basis are given by Eq. (\[E:eaa\]). Its diagonal matrix elements coincide with the $\varepsilon_h$, $\varepsilon_p$ energies defined in Eqs. (\[E:spvareh\]), (\[E:spvarep\]). It is possible to diagonalize the s.p. Hamiltonian in order to obtain its eigenvalues, $e_{\alpha}$.

All s.p. energies defined above have been formally evaluated in terms of matrix elements of the two-body Hamiltonian $H_{\text{int}} $. They should be further corrected for the center-of-mass energy, as was done in Ref. [@PPH2006] for the HF s.p. energies. For the purposes of this work, however, it will suffice to compare our uncorrected results with the uncorrected HF energies, in order to show the relative effect of RPA correlations.

One should keep in mind, that only for s.p. levels around the Fermi energy is it sensible to discuss discrete s.p. energies. For low-lying holes or high-lying particles the s.p. strength is strongly fragmented in realistic cases — mainly due to short-range correlations [@DiB2004].

Transition strength
-------------------

We consider the response of the nucleus to an external field described by the s.p. multipole operator $$O_{{J}^{\pi}{M}} = \sum_{\alpha m_{\alpha}  \beta m_{\beta}} 
\langle \alpha m_{\alpha} | O_{{J}^{\pi}{M}} 