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

TL;DR

This paper derives the residual interaction expressions for Second RPA with density-dependent forces, highlighting the importance of correctly including rearrangement terms, which differ between RPA and beyond RPA calculations.

Contribution

It provides a detailed derivation showing that the residual interaction and rearrangement terms differ between RPA and beyond RPA, correcting common assumptions.

Findings

Residual interaction expressions are derived for Second RPA with density-dependent forces.

Rearrangement terms differ between RPA and beyond RPA matrix elements.

Common prescriptions neglecting these differences are incorrect.

Abstract

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.