H_{\alpha\beta\alpha'\beta'} 
a_{\alpha}^{\dagger}a_{\beta}^{\dagger} 
a_{\beta'}a_{\alpha'} 
,
\label{E:ham}$$ containing a two-nucleon interaction and the $A$-dependent intrinsic part of the kinetic energy, as introduced in Sec. \[ScorrH\], Eq. (\[eq:hintr\]).

The remaining task is to evaluate the expectation values of the double commutators in Eqs. (\[E:Agen\]) and (\[E:Bgen\]). In general, elements of both the one-body and the two-body density matrices will appear in the commutator of the Hamiltonian $H_{\text{int}}$ with a $ph$ operator. The elements of the two-body density matrix can be tackled using a linearization procedure [@Bro1967; @Row1970], by which they are contracted with respect to a reference state (the ground state) so that a quantity linear in the $ph$ operators is obtained. Then, by substituting in Eqs. (\[E:Agen\]) and (\[E:Bgen\]), the matrix elements of $A$ and $B$ can be expressed in terms of the OBDM only. In the standard RPA, one uses the HF state (for which $\rho_h=1$, $\rho_p=0$) as the reference state. In the present ERPA the correlated ground state $|0\rangle$ is chosen as the reference state. By performing this linearization, one obtains $$\begin{aligned}
A_{php'h'}^{{J}} 
&=& \frac{1}{2}( 
D^{1/2}_{ph} D^{-1/2}_{p'h'} + 
D^{-1/2}_{ph} D^{1/2}_{p'h'}) 
\nonumber \\ 
& & \,\,\,\, \times  
(\delta_{hh'} \varepsilon_{p'p} - \delta_{pp'} \varepsilon_{hh'} )  
\nonumber \\ 
&& 
+ 
D^{1/2}_{ph} D^{1/2}_{p'h'} 
\langle p 