[@VKC2000], we call this formulation Extended RPA (ERPA) to distinguish it from the one which is based on the uncorrelated HF ground state.

Next, we review the ERPA method based on Refs. [@CPS1996; @CGP1998; @VKC2000] and we present the formalism used to evaluate ground-state properties and transition strengths. Expressions with explicit angular-momentum coupling are shown. Closed-shell nuclei are considered and spherical symmetry is assumed. The symbol $p(h)$ will represent all the quantum numbers of a particle (hole) state except the magnetic quantum number $m_p (m_h)$, i.e., the set of quantum numbers $\{n_{p(h)} \ell_{p(h)} j_{p(h)} t_{p(h)} \}$ of the nodes, orbital angular momentum, total angular momentum, and isospin. The letters $\alpha , \beta , ... $ will be used to denote single particle states of either kind ($p$ or $h$). As a single-particle basis, the one that diagonalizes the one-body density matrix (OBDM) is chosen — i.e., the natural-orbital basis. The ERPA formalism is based on this choice.

The ERPA equations
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The creation operator $Q^{\dagger}_{\nu ,{J}^{\pi}{M}}$ of an excited state $|\nu\rangle$ with multipolarity ${J}$ and parity $\pi $ is written as a linear combination of renormalized $ph$ creation and annihilation operators $\mathcal{A}_{ph}^{{JM} \dagger}$, $\mathcal{A}_{ph}^{{JM}}$, $$\begin{aligned}
|\nu\rangle &=& Q^{\dagger}_{\nu ,{J}^{\pi}{M}}| 0 