Van der Waals density functional: Self-consistent potential and the nature of the van der Waals bond

We derive the exchange-correlation potential corresponding to the nonlocal van der Waals density functional [M. Dion, H. Rydberg, E. Schröder, D. C. Langreth, and B. I. Lundqvist, Phys. Rev. Lett. 92, 246401 (2004)]. We use this potential for a self-consistent calculation of the ground state properties of a number of van der Waals complexes as well as crystalline silicon. For the latter, where little or no van der Waals interaction is expected, we find that the results are mostly determined by semilocal exchange and correlation as in standard generalized gradient approximations (GGA), with the fully nonlocal term giving little effect. On the other hand, our results for the van der Waals complexes show that the self-consistency has little effect on the atomic interaction energy and structure at equilibrium distances. This finding validates previous calculations with the same functional that treated the fully nonlocal term as a post-GGA perturbation. A comparison of our results with wave-function calculations demonstrates the usefulness of our approach. The exchange-correlation potential also allows us to calculate Hellmann-Feynman forces, hence providing the means for efficient geometry relaxations as well as unleashing the potential use of other standard techniques that depend on the self-consistent charge distribution. The nature of the van der Waals bond is discussed in terms of the self-consistent bonding charge.

Figure 1

(Color online) The kernels ${\Phi }_1$, ${\Phi }_2$, and ${\Phi }_3$ given in Eqs. (18) are plotted as a function of $D$ for values of $\delta =\pm 0.9$, $\pm 0.5$, and 0.0 [see Eqs. (19)].

Figure 2

(Top) Interaction energy of the Ar dimer as a function of separation. Plotted are self-consistent and non-self-consistent results. In addition, we show results where $E_c^{\mathrm{nl}}$ has been neglected. Experimental values are taken from Ref. 47. (Bottom) Forces calculated as the derivative of the energy $(-dE∕dr)$ and the Hellmann-Feynman forces.

Figure 3

Same as Fig. 2 except that the Kr dimer is described here.

Figure 4

Interaction energy of a $\mathrm{C}{\mathrm{O}}_2$ dimer as a function of its separation at a fixed slip distance of $2.05\mathrm{\AA }$. The inset shows the $\mathrm{C}{\mathrm{O}}_2$ dimer geometry.

Figure 5

(Color online) Interaction energy of water on benzene. The inset shows the geometry.

Figure 6

Interaction energy of a cytosine dimer as a function of separation.

Figure 7

Total energy (per two-atom unit cell) of Si as a function of the lattice constant. The inset shows the convergence of $E_c^{\mathrm{nl}}$ with respect to the cutoff radius $R_{\max }$. See text for further details.

Figure 8

(Color online) The bonding charge of a van der Waals complex. Shown is the difference in induced electron density $n_{\mathrm{ind}}^{\text{full}}-n_{\mathrm{ind}}^{\mathrm{no}\mathrm{nl}}$ for the Ar dimer with a separation of $4.02\mathrm{\AA }$. The scale is in Å and the black dots mark the position of the nuclei. The zero level is marked by the black contour. Red areas represent areas of electron density gain when the nonlocal part is included; conversely, blue areas indicate loss of electron density. Increments between contour lines are $5\times {10}^{-5}\text{electrons}∕{\mathrm{\AA }}^3$.