First-principles study of the binding energy between nanostructures and its scaling with system size

The equilibrium van der Waals binding energy is an important factor in the design of materials and devices. However, it presents great computational challenges for materials built up from nanostructures. Here we investigate the binding-energy scaling behavior from first-principles calculations. We show that the equilibrium binding energy per atom between identical nanostructures can scale up or down with nanostructure size, but can be parametrized for large $\mathcal{N}$ with an analytical formula (in meV/atom), ${E}_{b}/\mathcal{N}=a+b/\mathcal{N}+c/{\mathcal{N}}^{2}+d/{\mathcal{N}}^{3}$, where $\mathcal{N}$ is the number of atoms in a nanostructure and $a$, $b$, $c$, and $d$ are fitting parameters, depending on the properties of a nanostructure. The formula is consistent with a finite large-size limit of binding energy per atom. We find that there are two competing factors in the determination of the binding energy: Nonadditivities of van der Waals coefficients and center-to-center distance between nanostructures. To decode the detail, the nonadditivity of the static multipole polarizability is investigated from an accurate spherical-shell model. We find that the higher-order multipole polarizability displays ultrastrong intrinsic nonadditivity, no matter if the dipole polarizability is additive or not.