Random-phase approximation in a local representation

The random-phase approximation (RPA) in Q-P representation is introduced. It is shown that the RPA equations can be regained from varying the ratio of energy-weighted moments ${\mathit{m}}_{3}$(Q)/${\mathit{m}}_{1}$(Q) with respect to the particle-hole operator Q. In particular, a restricted set of local operators Q(r) is discussed, leading to a hydrodynamic approximation to the RPA. The practical solution of the collective eigenvalue problem for a given multipolarity proceeds via a power expansion of Q(r) and the solution of a secular equation for coupled modes. As an application of our method, we discuss collective dipole vibrations (plasmons) in small spherical metal clusters.