Insensitivity of the Infinite-Wavelength Surface Plasmon Frequency to the Electron Density Profile

Assuming that the ground-state electron density at a jellium-vacuum interface is the self-consistent Hartree density, and using the random-phase approximation to describe the electronic excitation spectrum, we show that the infinite-wavelength surface plasmon frequency is 1√2 times the bulk plasma frequency. This result is independent of the shape of the jellium background, and thus applies to a wide range of self-consistent electron density profiles.