Duality of the collective and single particle responses in simple metals in the extreme long wavelength limit

It is demonstrated that the collective and single particle responses of simple metals in the extreme long wavelength limit become identical. When applied to alternative models of a metallic slab, this duality proves equivalence of the plasma frequency and the surface energy barrier, which confines electrons inside the metal. This in-turn proves also a simple, yet powerful formula, which expresses the work function of a metals as a difference between its free-electron plasmon and Fermi energies, and which has been shown to be more accurate (even for complex metals) than the best available simulations.