Versatile van der Waals Density Functional Based on a Meta-Generalized Gradient Approximation

A “best-of-both-worlds” van der Waals (vdW) density functional is constructed, seamlessly supplementing the strongly constrained and appropriately normed (SCAN) meta-generalized gradient approximation for short- and intermediate-range interactions with the long-range vdW interaction from $r\mathrm{VV}10$, the revised Vydrov–van Voorhis nonlocal correlation functional. The resultant $\mathrm{SCAN}+r\mathrm{VV}10$ is the only vdW density functional to date that yields excellent interlayer binding energies and spacings, as well as intralayer lattice constants in 28 layered materials. Its versatility for various kinds of bonding is further demonstrated by its good performance for 22 interactions between molecules; the cohesive energies and lattice constants of 50 solids; the adsorption energy and distance of a benzene molecule on coinage-metal surfaces; the binding energy curves for graphene on Cu(111), Ni(111), and Co(0001) surfaces; and the rare-gas solids. We argue that a good semilocal approximation should (as SCAN does) capture the intermediate-range vdW through its exchange term. We have found an effective range of the vdW interaction between 8 and 16 Å for systems considered here, suggesting that this interaction is negligibly small at the larger distances where it reaches its asymptotic power-law decay.

Popular Summary

Atoms group together to form a range of materials via different types of chemical bonds: metallic, ionic, covalent, hydrogen, and van der Waals bonds. A theoretical tool able to describe all of these with reasonable accuracy and efficiency has been a long-term goal of both quantum physicists and chemists. The recently developed strongly constrained and appropriately normed (SCAN) meta-generalized-gradient-approximation (mGGA), within the framework of density functional theory, is a substantial step toward this goal. This approximation describes metallic, ionic, and covalent bonds very well and also captures intermediate-range van der Waals interactions. Here, we focus on describing nonlocal and long-range van der Waals interactions, which play important roles in sparse matter like molecular crystals and layered materials.

We select a nonlocal van der Waals correlation density functional to combine with the SCAN mGGA, opting to focus on the revised Vydrov–Van Voorhis functional because it is easy to implement, adaptable to other functionals, and well suited to molecular systems. The approximation that we recover possesses the desired versatility and works well for diverse problems in which the van der Waals interactions are either dominant or nearly negligible. We make use of benchmarking calculations of molecular complexes, physisorption of benzene on coinage metal surfaces, graphene on metal surfaces, metals, and covalent and ionic solids. We additionally find that our approximation demonstrates excellent accuracy simultaneously for layer-layer binding energy and spacing and the intralayer lattice constant for 28 layered materials, highlighting its capability to describe the van der Waals and non–van der Waals bonds on the same footing, particularly in near-equilibrium cases. Our approximation is also vastly more computationally efficient than previous methods of comparable accuracy.

We anticipate that our results will pave the way for theoretical studies of layered materials such as bilayers, few layers, and heterostructures.

Figure 1

The binding curves for Ar and Kr dimers from PBE, $r\mathrm{VV}10$, SCAN, and $\mathrm{SCAN}+r\mathrm{VV}10$ compared to CCSD(T) curves [40, 41] as the reference (REF).

Figure 2

Box plots for the absolute relative errors and relative errors of the interaction energies from PBE, $r\mathrm{VV}10$, vdW-DF2 (numerical results from Ref. [10]), SCAN, and $\mathrm{SCAN}+r\mathrm{VV}10$ with respect to the CCSD(T) results [37, 38], for the molecular dimers in the S22 data set. The PBE errors, too big to display on this scale, have been scaled down by a factor of 3 in this figure. The Tukey box plot used here summarizes the overall distribution of a set of data points: The bottom and the top of the box are the first (Q1) and third (Q3) quartiles (25% of data points lies below Q1, and another 25% above Q3); the band inside the box denotes the median; the circles, if any, denote outliers that lie further than $1.5*|\mathrm{Q}3-\mathrm{Q}1|$ away from the box; the vertical line extends from the minimum to the maximum, except for the outliers. We also denote the mean value with a shape inside the box.

Figure 3

Box plots for the absolute relative errors and relative errors of the atomization energies $E_a$, and the equilibrium lattice volumes $V_0$, from RPA, PBE, SCAN, and $\mathrm{SCAN}+r\mathrm{VV}10$ for 50 solids, with respect to the experimental values. The RPA, PBE, and experimental values are from Refs. [48, 49].

Figure 4

The calculated binding curves for a graphene sheet in registry with (a) Ni(111), (b) Co(0001), and (c) Cu(111) surfaces from PBE, SCAN, and $\mathrm{SCAN}+r\mathrm{VV}10$, compared with the RPA results [69].

Figure 5

Box plots for the absolute relative errors and relative errors of the interlayer binding energies, and interlayer and intralayer lattice constants ($c$ and $a$) from rev-vdW-DF2, SCAN, and $\mathrm{SCAN}+r\mathrm{VV}10$, for 28 layered materials. The reference values are from RPA for the binding energy and from experiment for the lattice constants [17, 19].