Incorrect asymptotics of dispersion energy in dispersion-augmented density functional approximations

The dispersion energies in dispersion-augmented density-functional (DFT+D) methods are usually determined using the supermolecular (SM) approach and the atoms-in-molecule approximation. These energies are computed from atom-distributed polarisabilities obtained by scaling some atom-assigned reference polarisabilities. The scaling may depend on the nature of its neighbouring atoms, their separation, and their orientation. Thus, the polarisability of an atom in a dimer is different from the polarisability of the same atom in an isolated monomer. In consequence, the dispersion contributions from pairs of atoms within a monomer do not cancel in the SM subtraction of monomer energies from dimer energies. This introduces a spurious nonphysical contribution that is present for all intermolecular separations R. In the van der Waals minimum region, this error is a negligible fraction of the total dispersion energy (except for DFT+D variants with weak damping), but at asymptotic separations, it may become several times larger than the true value of the dispersion energy. One reason for this dependence on R is that the contributions to dispersion energies are computed even between atoms separated by distances close to the chemical bond lengths, and such contributions, very large for all R, do not subtract completely in the SM method. We investigated several popular dispersion-correction functions and found that D4 suffers the most from this nonphysical behaviour, followed by XDM. For MBD the effect is small but still significant, while for D3 and VV10 it is essentially negligible. However, in the asymptotic region, this error becomes evident for all methods. The error is absent in approaches that are perturbative in nature. Some of the methods listed above can be used in perturbative way. If this is not possible, the error cannot be completely avoided, but it can be diminished by introducing a damping function that removes the contributions of atoms separated by chemical bond distances within a monomer.