Multipole response of metal spheres toq-dependent excitation operators

We have obtained the once and thrice energy-weighted moments of the random-phase-approximation (RPA) response to q-dependent excitation operators of type j L (qr) Y L 0 for metal spheres described within a spherical jellium model. These two moments, in conjunction with the Thomas-Fermi estimation of the RPA inverse energy-weighted moment, are used to study the response of these systems as a function of q. For small values of q, we recover the surface-mode systematics, whereas for large q’s the response is mainly determined by electron-hole excitations. For intermediate q values, bulk oscillations are found and their connection with the hydrodynamical-model predictions is established. In the limit of a big sphere, we have obtained an improved bulk-plasmon pole approximation for the dispersion relation which includes in a very easy way exchange and correlation effects. We have found that these corrections are not negligible. The moments of the response corresponding to a plane wave e i q ⋅ r are also discussed. Numerical applications to the case of Na spheres whose ground-state structure is described by models of different complexity (constant electronic density, Thomas-Fermi or Kohn-Sham) are presented.