  We show that the usual sum of $R^{-6}$ contributions from elements separated
by distance $R$ can give \emph{qualitatively} wrong results for the
electromagnetically non-retarded van der Waals interaction between
non-overlapping bodies. This occurs for anisotropic nanostructures that have a
zero electronic energy gap, such as nanowires, conducting nanotubes, and
nano-layered 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 measurable at sub-micron separations.
