  The DFT/vdW-QHO-WF method, recently developed to include the van der Waals
(vdW) interactions in approximated Density Functional Theory (DFT) by combining
the Quantum Harmonic Oscillator model with the Maximally Localized Wannier
Function technique, is applied to the cases of atoms and small molecules (X=Ar,
CO, H$_2$, H$_2$O) weakly interacting with benzene and with the ideal planar
graphene surface. Comparison is also presented with the results obtained by
other DFT vdW-corrected schemes, including PBE+D, vdW-DF, vdW-DF2, rVV10, and
by the simpler Local Density Approximation (LDA) and semilocal Generalized
Gradient Approximation (GGA) approaches. While for the X-benzene systems all
the considered vdW-corrected schemes perform reasonably well, it turns out that
an accurate description of the X-graphene interaction requires a proper
treatment of many-body contributions and of short-range screening effects, as
demonstrated by adopting an improved version of the DFT/vdW-QHO-WF method. We
also comment on the widespread attitude of relying on LDA to get a rough
description of weakly interacting systems.
