VdW in VASP and QEIt is clear that both VASP and QE have different flavors of theory implemented to deal with the vdW dispersion in solids and molecules.
Van der Waal interactions can be accounted for in VASP during a structure optimization process. Setting the tag LVDW to TRUE initiates this theory. The DFT-D2 mthod of Grimme is implemented in VASP. DFT-D2 is a semi-empirical simulation of the VdW interactions. Details can be found in J Comp Chem 27, 1787 (2006) The effect of these interactions is added to the convention Kohn-Sham DFT energy
E_DFT-D = EKS_DFT + E_VDW_disp
In the DFT-D2 method implemented in VASP, the van der Waal interactions have been optimized for each of the DFT functionals in use. Additional parameters that can be defined include VDW_RADIUS (cutoff radius for pair interactions), VDW_SCALING (global scaling factor), VDW_D (damping parameter), VDW_C6 (C6 parameters for each species in POSCAR) and VDW_R0 (R0 parameters for POSCAR)
An implementation of DFT-D2 is also available in QE by using the london=.true. flag in the system namelist of a pw.x run.
An alternative method to account for vdW interactions in QE is to set input_dft to vdw-dft. This is an improved LDA correlation functional which accounts for long range electronic interactions.