Origin of the size-dependence of the equilibrium van der Waals binding between nanostructures

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 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 2kth 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. The size dependences of other morphologies or bonding types lie between, as shown by sodium clusters.