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The status of different extensions of the Random Phase Approximation (RPA) is reviewed. The general framework is given within the Equation of Motion Method and the equivalent Green's function approach for the so-called Self-Consistent RPA (SCRPA). The role of the Pauli principle is analyzed. A comparison among various approaches to include Pauli correlations, in particular, renormalized RPA (r-RPA), is performed. The thermodynamic properties of nuclear matter are studied with several cluster approximations for the self-energy of the single-particle Dyson equation. More particle RPA's are shortly discussed with a particular attention to the alpha-particle condensate. Results obtained concerning the Three-level Lipkin, Hubbard and Picket Fence Models, respectively, are outlined. Extended second RPA (ESRPA) is presented.

Coupled equations for even and odd particle number correlation functions are set up via the equation of motion method. For the even particle number case this leads to self-consistent RPA (SCRPA) equations already known from the literature. From the equations of the odd particle number case the single particle occupation probabilities are obtained in a self-consistent way. This is the essential new procedure of this work. Both, even and odd particle number cases are based on the same correlated vacuum and, thus, are coupled equations. Applications to the Lipkin model and the 1D Hubbard model give very good results., Comment: 12 pages, 27 figures, to be published in PRC. arXiv admin note: substantial text overlap with arXiv:1905.04672

Self-Consistent RPA is rederived in a consistent way with the help of the Coupled Cluster ground state wave function truncated at the two body level. An exact killing operator for this wave function is introduced allowing for a detailed discussion of the approximation scheme. Several exactly solvable models are reanalyzed under this new perspective giving raise to a quantitative evaluation of the performances of this many body method., Comment: 18 pages, 14 figures