Tkatchenko-Scheffler van der Waals correction method with and without self-consistent screening applied to solids

The method proposed by Tkatchenko and Scheffler [Phys. Rev. Lett. 102, 073005 (2009)] to correct density functional calculations for the missing van der Waals interactions is implemented in the Vienna ab initio simulation package (vasp) code and tested on a wide range of solids, including noble-gas crystals, molecular crystals ($\alpha$-N${}_2$, sulfur dioxide, benzene, naphthalene, cytosine), layered solids (graphite, hexagonal boron nitride, vanadium pentoxide, MoS${}_2$, NbSe${}_2$), chain-like structures (selenium, tellurium, cellulose I), ionic crystals (NaCl, KI), and metals (nickel, zinc, cadmium). In addition to the original formulation expressing the van der Waals (vdW) corrections as pairwise potentials whose strength is derived from the rescaled polarizabilities of the neutral free atoms, the self-consistently screened ($\text{TS}+\text{SCS}$) [Phys. Rev. Lett. 108, 236402 (2012)] variant of the method involving electrodynamic response effects has been examined. Analytical expressions for the forces acting on the atoms and for the components of the stress tensor needed for the relaxation of the volume and shape of the unit cell using the $\text{TS}+\text{SCS}$ method are derived. While the calculated structures are reasonably close to experiment, the van der Waals corrections to the binding energies are often found to be overestimated in comparison with experimental data. The $\text{TS}+\text{SCS}$ approach leads to significantly better results in some problematic cases, such as the binding energy of graphite. However, there is room for further improvements, in particular for strongly ionic systems.

Figure 1

Energy versus volume (above) and internal pressure versus volume (below) curves computed for NaCl using the $\text{TS}+\text{SCS}$ method with and without gradient corrections. The vertical dashed line marks the minimum in the total energy which must coincide with the volume for which the internal pressure vanishes.

Figure 2

Dispersion coefficient of carbon in graphite computed using PBE-TS and PBE-$\text{TS}+\text{SCS}$ as a function of interlayer separation $d$.