Correcting long-range electrostatics in DFTB

We demonstrate that the atom-based charge model implemented in the current versions of the density functional tight binding (DFTB) method fails to reproduce the correct charge distribution of a range of systems, including homonuclear molecules, graphene, and nanotubes, resulting in serious distortions in the electrostatic interactions for such systems caused by the missing quadrupole moments. In particular, this failure seriously impacts the long- and medium-range interaction energies of the DFTB plus dispersion (DFTB-D) model, leading to incorrect predictions of translational or rotational barriers in such systems. We show explicitly on examples of H₂ and N₂ that correct quadrupole moments - and consequently correct electrostatic interactions - can be restored in such systems by adding additional bond (ghost) sites to the homonuclear molecules. Attempts to determine the point charges associated with the additional sites using the usual Mulliken population analysis lead to unphysical results. Instead, these charges can be determined using the actual DFTB densities used in the parameterization process. For homonuclear molecules, we propose an extension to the DFTB-D model by adding charges that reproduce the physically correct quadrupolar charge distribution. The resulting DFTB-D-Q model greatly improves the rotational barriers for interactions of molecular hydrogen and nitrogen with benzene.