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We derive the expression for the residual interaction to be used in theframework of the Second RPA with density-dependent forces. The adoptedprocedure is based on a variational approach. It is found that the residualinteraction to be used in RPA and beyond RPA matrix elements is not the same asfar as the rearrangement terms are concerned. A detailed derivation of thematrix elements coupling 1 particle-1 hole with 2 particle-2 hole and 2particle-2 hole among themselves has been done and the correspondingrearrangement terms are shown here. This formal result indicates that both thecurrently used prescriptions, namely (i) using the same type of rearrangementterms in RPA and beyond RPA matrix elements or (ii) neglecting therearrangement terms in beyond RPA matrix elements, are not correct.

Residual interaction in Second RPA with density-dependent forces. Rearrangement terms