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The DFT/vdW-QHO-WF method, recently developed to include the van der Waals(vdW) interactions in approximated Density Functional Theory (DFT) by combiningthe Quantum Harmonic Oscillator model with the Maximally Localized WannierFunction 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 planargraphene surface. Comparison is also presented with the results obtained byother DFT vdW-corrected schemes, including PBE+D, vdW-DF, vdW-DF2, rVV10, andby the simpler Local Density Approximation (LDA) and semilocal GeneralizedGradient Approximation (GGA) approaches. While for the X-benzene systems allthe considered vdW-corrected schemes perform reasonably well, it turns out thatan accurate description of the X-graphene interaction requires a propertreatment of many-body contributions and of short-range screening effects, asdemonstrated by adopting an improved version of the DFT/vdW-QHO-WF method. Wealso comment on the widespread attitude of relying on LDA to get a roughdescription of weakly interacting systems.

Including screening in van-der-Waals corrected DFT calculations. The case of atoms and small molecules physisorbed on graphene