Interpretation of van der Waals density functionals

The nonlocal correlation energy in the van der Waals density functional (vdW-DF) method can be interpreted in terms of a coupling of zero-point energies of characteristic modes of semilocal exchange-correlation (xc) holes. These xc holes reflect the internal functional in the framework of the vdW-DF method. We explore the internal xc hole components, showing that they share properties with those of the generalized-gradient approximation. We use these results to illustrate the nonlocality in the vdW-DF description and analyze the vdW-DF formulation of nonlocal correlation.

Figure 1

Schematics of a typical problem where vdW-DF is called upon to describe the material binding: a system with multiple molecular-type regions that couple electrodynamically across internal “voids” with sparse electron distribution, Ref. [41]. Each molecular-type fragment can be described by a GGA-type description and such descriptions form the starting point for the vdW-DF evaluation of nonlocal correlations. The voids are not necessarily free of electrons but are regions in which there is only a tail (or overlapping tails) of the molecular-type electron distributions. The schematics is adapted from Fig. 7 of Ref. [37] and shows the atomic configuration and contours of the electron-density distribution of the molecular building blocks for a benzene molecule and a graphene sheet at the vdW-DF binding separation. We note that a natural delineation surface of minimum electron distribution (here illustrated by a dashed curve) runs through interfragment positions with a saddle-point or troughlike behavior in the electron concentration.

Figure 2

Contours of the scaled density gradients $s=\left|\nabla n\right|/\left(2k_{\mathrm{F}}n\right)$ in a benzene dimer at binding separation. The saddle-point behavior, at low-density and low-to-moderate $s$ values in the region between the molecules, is typical of binding with a significant vdW component, Refs. [37, 38].

Figure 3

Correlation (top) and exchange (bottom ) related components of the vdW-DF1 and vdW-DF2 internal functional enhancement factor $f_{\mathrm{xc}}^{\text{in}}=f_c^{\mathrm{LDA}}+f_x^{\text{in}}$. The first is a function of the density $n$ [or, equivalently, of $r_s={(3/4\pi /n)}^{1/3}$]. The second is a function of the scaled density gradient $s=\left|\nabla n\right|/\left(2k_{\mathrm{F}}n\right)$.

Figure 4

Contour plot of the vdW-DF internal functional xc hole ${\overline{n}}_{\mathrm{xc}}^{\mathrm{in}}$, spatially weighted and scaled according to Eq. (37). The hole is mapped as a function of scaled separation $z=2k_F|{\mathbf{r}}^{\prime }-\mathbf{r}|$ and the characteristic internal functional enhancement factor $f\left(\mathbf{r}\right)=f_{\mathrm{xc}}^0\left(\mathbf{r}\right)=q_0\left[n\right]\left(\mathbf{r}\right)/k_F\left[n\right]\left(\mathbf{r}\right)$, which is fixed for a given density $n\left(\mathbf{r}\right)$ and a given scaled density gradient $s=\left|\nabla n\right|/\left(2k_{\mathrm{F}}n\right)$. Contour spacing is 0.025.

Figure 5

Comparison in the homogeneous limit between the vdW-DF representation of the internal xc hole (top panel) and the Perdew and Wang [76] xc models (mid and bottom panels). Two electron densities, as specified by value of $r_s={(3/4\pi /n)}^{1/3}$ are considered. The middle panel relies on the exact exchange hole while the lower panel relies on a nonoscillatory approximation.

Figure 6

A comparison of the vdW-DF2 internal xc hole and a numerical xc hole consisting of exchange at the GGA level and correlation at the LDA level. The scaled gradient is chosen as $s=0.75$ and $1.5$ for two different densities. The hole deepens and narrows as $s$ increases.