h^{-1} ;{J}  |  H_{\text{int}}  |  p'{h'}^{-1} ; {J} \rangle  
\label{E:Amat} 
\\ 
B_{php'h'}^{{J}} 
&=& 
[D_{ph} D_{p'h'} 
(1+\delta_{hh'})(1+\delta_{pp'})]^{1/2} 
\nonumber \\ 
& & \,\,\,\, \times  
\langle (p h^{-1} ;{J}) (p'{h'}^{-1} ; {J}^{-1})  |  H_{\text{int}}  |  0 \rangle  
\label{E:Bmat} \end{aligned}$$ where $ | \alpha \beta ; J \rangle $ are the normalized and antisymmetrized, angular-momentum–coupled two-body states and $$\varepsilon_{\alpha \alpha '} \equiv 
\sum_{\beta ,J} \rho_{\beta}  
 [(1+\delta_{\alpha\beta})(1+\delta_{\alpha '\beta})]^{1/2} 
\langle \alpha \beta ;J  |  H_{\text{int}}  |  \alpha' \beta  ; J \rangle  
\label{E:eaa}$$ are matrix elements of the single-particle Hamiltonian (see Sec. \[S:gsp\]). The quantity $D_{ph}$ has been defined in Eq. (\[E:Nmat\]). For $\rho_h=1$, $\rho_p=0$ the usual RPA equations are recovered.

According to Eqs. (\[E:Amat\]) and (\[E:Bmat\]), the ERPA matrix (\[E:ERPAeq\]) depends on the occupation probabilities $\rho_{\alpha}$. These, as we will see, depend on the forward and backward amplitudes $X$ and $Y$ of all the ${J}^{\pi}$ states of the nucleus, i.e., on the solutions of the ERPA equations. Therefore, the equations must be solved iteratively. Let us call this procedure the ERPA iteration. The occupation probabilities of the single-particle states can be expressed in terms of the amplitudes $X$ and $Y$ and the matrix $N$ of Eq. (\[E:Nmat\]), using the number-operator method [@CPS1996]. Contrary to the quasi-boson 