  Nanostructures can be bound together at equilibrium by the van der Waals
(vdW) effect, a small but ubiquitous many-body attraction that presents
challenges to density functional theory. How does the binding energy depend
upon the size or number of atoms in one of a pair of identical nanostructures?
To answer this question, we treat each nanostructure properly as a whole
object, not as a collection of atoms. Our calculations start from an accurate
static dipole polarizability for each considered nanostructure, and an accurate
equilibrium center-to-center distance for the pair (the latter from experiment,
or from the vdW-DF-cx functional). We consider the competition in each term
$-C_{2k}/d^{2k}$ ($k=3, 4, 5$) of the long-range vdW series for the interaction
energy, between the size dependence of the vdW coefficient $C_{2k}$ and that of
the $2k$-th power of the center-to-center distance $d$. The damping of these
vdW terms can be negligible, but in any case it does not affect the size
dependence for a given term in the absence of non-vdW binding. To our surprise,
the vdW energy can be size-independent for quasi-spherical nanoclusters bound
to one another by vdW interaction, even with strong nonadditivity of the vdW
coefficient, as demonstrated for fullerenes. We also show that, for
low-dimensional systems, the vdW interaction yields the strongest
size-dependence, in stark contrast to that of fullerenes. We illustrate this
with parallel planar polycyclic aromatic hydrocarbons. Other cases are between,
as shown by sodium clusters.
