(1 for holes and 0 for particles). Then we solve the eigenvalue problem of Eq. (\[E:ERPAeq\]). At this stage, we have simply solved conventional RPA equations. Then we calculate a “new" OBDM by using Eqs. (\[E:OBDMp\]) and (\[E:OBDMh\]) and, as input, the $X$ and $Y$ amplitudes and the HF occupation probabilities. We proceed with a OBDM iteration and obtain new occupation probabilities, which hereafter replace the initial ones (the HF ones in this first step) and are used as input to build again the ERPA matrix for each ${J}^{\pi}$. Here begins the second ERPA-iteration step. We proceed as before, until convergence is reached.

Ground-state properties in the ERPA method {#S:gsp}
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The ERPA method provides information on the correlated RPA ground state. As we have seen, the natural orbitals can be determined and their occupation probabilities can be calculated up to fourth order in the backward amplitude. As a convenient way to quantify the ground-state correlations we will use the mean square deviation per particle of the OBDM [@AHP1993], defined by $$\sigma=\text{Tr}[(\varrho - \varrho^{(0)})^2] / A  
.
\label{E:sigma}$$ This quantity characterizes the deviation of the OBDM $\varrho$ of the correlated ground state from the OBDM $\varrho^{(0)}$ of the uncorrelated state 