Performance of the strongly constrained and appropriately normed density functional for solid-state materials

Constructed to satisfy 17 known exact constraints for a semilocal density functional, the strongly constrained and appropriately normed (SCAN) meta-generalized-gradient-approximation functional has shown early promise for accurately describing the electronic structure of molecules and solids. One open question is how well SCAN predicts the formation energy, a key quantity for describing the thermodynamic stability of solid-state compounds. To answer this question, we perform an extensive benchmark of SCAN by computing the formation energies for a diverse group of nearly 1000 crystalline compounds for which experimental values are known. Due to an enhanced exchange interaction in the covalent bonding regime, SCAN substantially decreases the formation energy errors for strongly bound compounds, by approximately 50% to 110 meV/atom, as compared to the generalized gradient approximation of Perdew, Burke, and Ernzerhof (PBE). However, for intermetallic compounds, SCAN performs moderately worse than PBE with an increase in formation energy error of approximately 20%, stemming from SCAN's distinct behavior in the weak bonding regime. The formation energy errors can be further reduced via elemental chemical potential fitting. We find that SCAN leads to significantly more accurate predicted crystal volumes, moderately enhanced magnetism, and mildly improved band gaps as compared to PBE. Overall, SCAN represents a significant improvement in accurately describing the thermodynamics of strongly bound compounds.

Figure 1

Jacob's ladder framework for describing different levels of sophistication and accuracy of $E_{\text{xc}}$ in density functional theory. Starting from the Hartree level of theory ($E_{\text{xc}}=0$) and attempting to climb towards chemical accuracy, each rung of the ladder labeled on the left corresponds to additional dependencies of $E_{\text{xc}}$ indicated on the right. The top rung listed is the random phase approximation (RPA). Figure is based on Refs. [3, 4]. We note that additional methodologies have been described as rungs between meta- and hyper-GGA not shown here [5, 6].

Figure 2

Non-spin-polarized exchange enhancement factor for the LDA, PBE, and SCAN density functionals as a function of dimensionless density gradient for different $\alpha$. The SCAN functional is constructed based on three different limits of $\alpha$ corresponding to the different bonding regimes indicated.

Figure 3

Comparison of calculated and experimental experimental formation energy for the 945 compounds for PBE and SCAN. Multiple points for the same compound and functional correspond to different sources of experimental formation energy. The dashed diagonal line corresponds to the $\mathrm{\Delta }{H}_{\mathrm{calc}}=\mathrm{\Delta }{H}_{\mathrm{expt}}$ line of perfect agreement. Each bar chart on the axes corresponds to a histogram, which is stacked in the case of the experimental formation energy.

Figure 4

(a) Absolute and (b) relative errors of the calculated formation energies plotted against the experimental formation energies. The dashed vertical lines correspond to the $\mathrm{\Delta }{H}_{\mathrm{calc}}=\mathrm{\Delta }{H}_{\mathrm{expt}}$ line of perfect agreement. Histograms of the errors are shown at the top. For the relative errors, the range is limited to $\pm 150\%$.

Figure 5

Comparison of absolute and relative errors for PBE and SCAN for (a) 297 strongly bound compounds, (b) 648 weakly bound compounds, and (c) all 945 compounds. Absolute (relative) errors are plotted with respect to the left (right) axis.

Figure 6

(a) Density-weighted probability distributions of $r_s$, $s$, and $\alpha$ in real space for CaO and three intermetallic compounds. The distributions are normalized to unity. (b) The compound $\alpha$ formation distribution defined in analogy to the formation energy of Eq. (4). (c) Non-spin-polarized exchange-correlation enhancement factor at $r_s=1$ bohr for LDA, PBE, and SCAN at several relevant $\alpha$ values.

Figure 7

Elemental chemical potential corrections $\delta \mu$ obtained via fitting for (a) PBE and (b) SCAN. Gray-colored squares correspond to elements not considered in the compound set. The training and testing $\mathrm{\Delta }E$ RMSE from ninefold cross validation are shown in (c) for the cases of fitting $\mu$ for no (“fit-none”) and all (“fit-all”) elements.

Figure 8

Comparison of predicted structural, magnetic, and electronic properties for PBE and SCAN, with experimental values shown when available. (a) Absolute and relative errors in compound volume. (b) Spontaneous magnetization for Fe, Co, and Ni. (c) Average maximum local magnetic moment for magnetic compounds. (d) Calculated and experimental electronic band gap, with the dashed line corresponding to perfect agreement.