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

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