Asymptotics of the Dispersion Interaction: Analytic Benchmarks for van der Waals Energy Functionals

We show that the usual sum of $R^{-6}$ contributions from elements separated by distance $R$ can give qualitatively wrong results for the electromagnetically nonretarded van der Waals interaction between nonoverlapping bodies. This occurs for anisotropic nanostructures that have a zero electronic energy gap, such as metallic nanotubes or nanowires, and nanolayered systems including metals and graphene planes. In all these cases our analytic microscopic calculations give an interaction falling off with a power of separation different from the conventional value. We discuss implications for van der Waals energy functionals. The new nanotube interaction might be directly observable at submicron separations.