Rehabilitation of the Perdew-Burke-Ernzerhof generalized gradient approximation for layered materials

The structural and energetic properties of layered materials present a challenge to density functional theory with common semilocal approximations to the exchange-correlation energy. By combining the most widely used semilocal generalized gradient approximation (GGA), the Perdew-Burke-Ernzerhof (PBE) one, with the revised Vydrov–van Voorhis nonlocal correlation functional (rVV10), both excellent structural and energetic properties of 28 layered materials have been recovered with a judicious parameter selection. We term the resulting functional PBE+rVV10L, with the “L” indicating that it is for layered materials. Such a combination is not new, and only involves refitting a single global parameter. However, the resulting excellent accuracy suggests such a dispersion-corrected PBE for many aspects of theoretical studies on layered materials. For comparison, we also present the results for PBE+rVV10 where the parameter is determined by 22 interaction energies between molecules.

Figure 1

Box plots for the absolute relative errors and relative errors of the interlayer binding energies ($E_b$), inter- and intralayer lattice constants ($c$ and $a$) from SCAN+rVV10 [21], PBE+rVV10, and PBE+rVV10L, for 28 layered materials. The reference values are from RPA for the binding energy, and from experiment for the lattice constants [17, 19]. 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 which lie further than $1.5*|Q3-Q1|$ away from the box. The vertical line extends from the minimum to the maximum, except for the outliers. Besides, we also denote the mean value with a shape inside the box.

Figure 2

Box plots for the absolute relative errors and relative errors of the atomization energies $E_a$, and the lattice equilibrium volumes $V_0$, from RPA, PBE, PBE+rVV10, and $\mathrm{PBE}+r\mathrm{VV}10\mathrm{L}$ for 50 solids, with respect to the experimental values. The RPA, PBE, and experimental values (after the zero-point correction) are from Refs. [39, 40].

Figure 3

Box plots for the absolute relative errors and relative errors of the interaction energies from $r\mathrm{VV}10$, SCAN+rVV10, PBE+rVV10, PBE+rVV10L, with respect to the CCSD(T) results [26, 27], for the molecular complexes in the S22 data set.