Long-range interaction of anisotropic systems

The first-order electrostatic interaction energy between two far-apart anisotropic atoms depends not only on the distance between them but also on their relative orientation, according to Rayleigh Schr\"{o}dinger perturbation theory. Using the first-order interaction energy and the continuum model, we study the long-range interaction between a pair of parallel graphene sheets. The asymptotic form of the obtained potential density, $ \varepsilon(D) \propto -D^{-3}-O(D^{-4})$, is consistent with the random phase approximation and Lifshitz theory. Accordingly, neglectance of the anisotropy, especially the nonzero first-order interaction energy, is the reason that the widely used Lennard-Jones potential approach and dispersion-corrections in density functional theory give a wrong asymptotic form, $ \varepsilon(D) \propto -D^{-4}$.