Van der Waals density functionals applied to solids

The van der Waals density functional (vdW-DF) of M. Dion et al. [Phys. Rev. Lett. 92, 246401 (2004)] is a promising approach for including dispersion in approximate density functional theory exchange-correlation functionals. Indeed, an improved description of systems held by dispersion forces has been demonstrated in the literature. However, despite many applications, standard general tests on a broad range of materials including traditional “hard” matter such as metals, ionic compounds, and insulators are lacking. Such tests are important not least because many of the applications of the vdW-DF method focus on the adsorption of atoms and molecules on the surfaces of solids. Here we calculate the lattice constants, bulk moduli, and atomization energies for a range of solids using the original vdW-DF and several of its offspring. We find that the original vdW-DF overestimates lattice constants in a similar manner to how it overestimates binding distances for gas-phase dimers. However, some of the modified vdW functionals lead to average errors which are similar to those of PBE or better. Likewise, atomization energies that are slightly better than from PBE are obtained from the modified vdW-DFs. Although the tests reported here are for hard solids, not normally materials for which dispersion forces are thought to be important, we find a systematic improvement in cohesive properties for the alkali metals and alkali halides when nonlocal correlations are accounted for.

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Figure 1

Comparison of the relative errors in the lattice constants calculated with different vdW-DF functionals, LDA (from Ref. 41), PBE, PBEsol, and recent data using the RPA (Ref. 51). The vdW methods are shown in the left panel while the right panel contains the results of the (semi)local functionals and RPA, as well as the optB86b-vdW results for comparison. All the methods overestimate the lattice constants for ionic solids and semiconductors and tend to give shorter lattices for alkali and alkali-earth metals. As with the S22 set, both the revPBE-vdW and rPW86-vdW2 yield equilibrium distances that are too long in most cases. This is improved by the functionals with optimized exchange: optPBE-vdW, optB88-vdW, and optB86b-vdW.

Figure 2

The exchange enhancement factors $F_x$ of the functionals employed in this study: PBE, PBEsol, and revPBE, which share the same functional form but differ in the values of parameters; rPW86, which is used in the rPW86-vdW2 (Ref. 21) functional; and three exchange functionals (optPBE, optB88, and optB86b) optimized for use with the vdW correlation (Ref. 15). The steepness for small reduced density gradients ($s$) is of crucial importance in determining the lattice constants.

Figure 3

Relative errors in the bulk moduli for the methods considered in this study. Results of the vdW functionals are shown on the left, results of the other methods on the right, where also the optB86b-vdW results were added. The original revPBE-vdW and rPW86-vdW2 methods underestimate the moduli by up to 45% and the optimized vdW functionals reduce the errors.

Figure 4

Relative errors in atomization energies calculated using different DFT approaches. Data for various flavors of vdW functionals are shown in the left panel. Data for LDA, semilocal PBE and PBEsol, the RPA method, and the optB86b-vdW functional are shown in the right panel. The ZPE was subtracted from the experimental data. The optB88-vdW and optB86b-vdW tend to give values between those of PBE and PBEsol for transition metals and semiconductors, in agreement with the behavior of their exchange enhancement factor. However, they agree better with the reference for alkali halides, where even PBEsol underbinds. Moreover, they give consistent errors for alkali metals where both PBE and PBEsol increasingly underbind with the increasing size of the ion.

Figure 5

Comparison of the relative errors in the lattice constants for the PBE exchange functional with LDA, PBE, and vdW correlation functionals.

Figure 6

Relative errors in the atomization energies for the PBE exchange functional combined with LDA, PBE, and vdW correlation functionals.

Figure 7

Lattice constants of various solids calculated with different approximations of the nonlocal van der Waals energy for the optB86b-vdW functional. The self-consistent implementation in VASP and non-self-consistent calculations based on the same density are reported. These use the real valence density (“valence”) and the all-electron density without (“AE no soft”) and with (“AE dens. cut”) the soft correction. The VASP calculations tend to give better agreement with the AE calculations because of the partial electronic core charge density added to the pseudo–valence density.