described by a Slater determinant $|\Psi^{(0)}\rangle$. It describes how well $|\Psi^{(0)}\rangle$ approximates the correlated state. It is minimal when evaluated in the natural-orbital basis, with $|\Psi^{(0)}\rangle$ built of the $A$ hole states [@Kob1969]. Then $$\sigma = \sigma_{\min}=\frac{1}{A} 
\left[ 
\sum_h \hat{j}_h^2 (1-\rho_h)^2 
+ \sum_p \hat{j}_p^2   \rho_p^2 
\right] 
\label{E:sigmaNO}$$ expresses the depletion of the hole states and the occupation of the particle states as an average. In HF theory $\sigma_{\min}$ obviously vanishes, as does the Fermi-sea depletion, while in realistic cases $\sigma_{\min}$ is expected to take values of the order $10^{-2}$ [@AHP1993; @JMN1987], corresponding to a Fermi-sea depletion of at least 10%. It has long been observed [@JMN1987] that values way below $10^{-2}$ indicate that short- and medium-range correlations have not been (adequately) taken into account by the model in question. This is typically the case, e.g., in RPA calculations. We will return to this point in Sec. \[Sresults\].

Removal and pick-up centroid energies, $\varepsilon^{(-)}$ and $\varepsilon^{(+)}$, respectively, can also be defined [@Row1968a; @Row1970; @Fri1975] and evaluated within the ERPA. The removal energy $\varepsilon^{(-)}_h$ corresponding to the hole state $h$ is the centroid of the distribution $$\begin{aligned}
S^{(-)}_h (E) &=& \sum_{fm_h} |\langle f | a_{hm_h} | 0 \rangle |^2 \delta (E-E_f)  
  \nonumber 