$X$ and $Y$ of Eq. (\[E:nu\]) obey the orthonormalization condition $$\sum_{ph} ( 
X_{ph}^{\nu ' , {J}^{\pi}            \ast} X_{ph}^{\nu , {J}^{\pi}           } 
- 
Y_{ph}^{\nu ' , {J}^{\pi}            \ast} Y_{ph}^{\nu , {J}^{\pi}           } 
) 
=\delta_{\nu\nu '} 
.$$

The equations determining the amplitudes $X$ and $Y$ are obtained by using the equations-of-motion method [@Row1968a]. As usual, they are written as a matrix equation in the $ph$ space, $$\left( 
\begin{array}{cc}  
A & B \\ -B^{\ast} & -A^{\ast} 
\end{array} 
\right) 
\left( 
\begin{array}{c} 
X^{\nu} 
\\ 
Y^{\nu} 
\end{array} 
\right) 
= E_{\nu}  
\left( 
\begin{array}{c} 
X^{\nu} 
\\ 
Y^{\nu} 
\end{array} 
\right) 
, 
\label{E:ERPAeq}$$ where $E_{\nu}$ is the excitation energy of the state $|\nu\rangle $ and the quantum numbers ${J}^{\pi}$ are implicit. The elements of the matrices $A$ and $B$, which are independent of ${M}$, are given by $$\label{E:Agen} 
A_{php'h'}^{{J}} = 
\langle 0 | [\mathcal{A}_{ph}^{{JM}}, \mathcal{H} ,\mathcal{A}_{p'h'}^{{JM} \dagger }] | 0 \rangle$$ and $$\label{E:Bgen} 
B_{php'h'}^{{J}} = 
-\langle 0 | [\mathcal{A}_{ph}^{{JM}}, \mathcal{H} ,(-1)^{{J}+{M}} 
\mathcal{A}_{p'h'}^{{J-M}}] | 0 \rangle  
,$$ where $\mathcal{H}$ is the Hamiltonian of the system. The symmetrized double commutators are defined as $$[A,B,C] = \frac{1}{2} ( [A,[B,C] ] + [ [A,B], C] ) 
.$$ Here we consider the two-body Hamiltonian, $$\mathcal{H} = H_{\text{int}} = \frac{1}{4} \sum_{\alpha\beta \alpha ' \beta ' } 
