Without doing much work, one can guess that the electron-electron interactions will scale like the inverse of the average electron-electron separation and hence as 1 / r 12 (since the Coulomb interaction goes like one over distance between charges) so that if we view the interactions as a small correction to the kinetic energy, we are describing the limit of small r s (i.e. 1 / r s 2 being larger than 1 / r s ) and hence high electron density. Unfortunately, real metals typically have r s between 2-5 which means this picture needs serious revision.