Nonuniversality of the dispersion interaction: analytic benchmarks for van der Waals energy functionals for soft layered and striated systems

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 and for experiment. We also discuss evidence for failures of standard theory for these systems near to their equilibrium spacing. THIS IS A MAJOR REVISION OF AN EARLIER VERSION, COND-MAT/0502422, BUT EACH VERSION CONTAINS SIGNIFICANT MATERIAL NOT FOUND IN THE OTHER, SO WE NEED THE TWO VERSIONS TO BE SEPARATELY CITEABLE.