van der Waals density functional with corrected $C_6$ coefficients

The non-local van der Waals density functional (vdW-DF) has had tremendous success since its inception in 2004 due to its constraint-based formalism that is rigorously derived from a many-body starting point. However, while vdW-DF can describe binding energies and structures for van der Waals complexes and mixed systems with good accuracy, one long-standing criticism---also since its inception---has been that the $C_6$ coefficients that derive from the vdW-DF framework are largely inaccurate and can be wrong by more than a factor of two. It has long been thought that this failure to describe the $C_6$ coefficients is a conceptual flaw of the underlying plasmon framework used to derive vdW-DF. We prove here that this is not the case and that accurate $C_6$ coefficient can be obtained without sacrificing the accuracy at binding separations from a modified framework that is fully consistent with the constraints and design philosophy of the original vdW-DF formulation. Our design exploits a degree of freedom in the plasmon-dispersion model $\omega_{\mathbf{q}}$, modifying the strength of the long-range van der Waals interaction and the cross-over from long to short separations, with additional parameters tuned_ to reference systems. Testing the new formulation for a range of different systems, we not only confirm the greatly improved description of $C_6$ coefficients, but we also find excellent performance for molecular dimers and other systems. The importance of this development is not necessarily that particular aspects such as $C_6$ coefficients or binding energies are improved, but rather that our finding opens the door for further conceptual developments of an entirely unexplored direction within the exact same constrained-based non-local framework that made vdW-DF so successful in the first place.