High-throughput hybrid-functional DFT calculations of bandgaps and formation energies and multifidelity learning with uncertainty quantification

Despite the fact that first-principles methods are critical tools in the study and design of materials today, the accuracy of density functional theory (DFT) prediction is fundamentally reliant on the exchange-correlation functional chosen to approximate the interactions between electrons. Although the general improvement in accurately calculating the bandgap with the Heyd-Scuseria-Ernzerhof (HSE) hybrid-functional method over the conventional semilocal DFT is well accepted, other properties such as formation energy have not been systematically studied and have yet to be evaluated thoroughly for different classes of materials. A high-throughput hybrid-functional DFT investigation on materials bandgaps and formation energies is therefore performed in this work. By evaluating over a thousand materials, including metals, semiconductors, and insulators, we have quantitatively verified that the materials bandgaps obtained through HSE [mean absolute error (MAE) = 0.687 eV] are more accurate than those from the Perdew-Burke-Ernzerhof (PBE) functional (MAE = 1.184 eV) when compared to the experimental values. For formation energies, PBE systematically underestimates the magnitude of the formation enthalpies (MAE = 0.175 eV/atom), whereas formation enthalpies obtained from the HSE method are generally more accurate (MAE = 0.147 eV/atom). We have also found that HSE significantly increases the accuracy of formation energy prediction for insulators and strongly bound compounds. A primary application of this new dataset is achieved by building a cokriging multifidelity machine learning (ML) model to quickly predict the bandgaps with HSE-level accuracy when its PBE bandgap is available from DFT calculations. The preliminary goal of our ML model, benchmarked in this work, is to select the semiconductors and insulators which may have been mislabeled as metals from the DFT-PBE calculations in the existing Open Quantum Materials Database. The performance of the cokriging model in reliably predicting HSE bandgaps with quantified model uncertainty is analyzed by comparing the results against published experimental data from the literature.

Figure 1

The workflow for hybrid-functional calculations implemented within the qmpy package. For given input structure(s), qmpy generates the vasp input files to perform a relaxation run with the PBE functional, followed by a calculation to generate the electronic wave function (wfc) and, finally, the hybrid-functional calculation. The taskserver and jobserver modules communicate with High-Performance Computing Cluster (HPCC) resources, monitor running jobs, and transfer data between local database storage and HPCC servers. The final calculation data undergo thermodynamic analysis and are stored in a MySQL database. The web interface of the Open Quantum Materials Database (OQMD) [26] provides convenient database access to the research community.

Figure 2

Comparison of bandgaps for 1135 compounds between PBE and HSE.

Figure 3

Comparisons between DFT (left: PBE, right: HSE) and experimental bandgaps for 146 compounds. The experimental data are taken from the Database of Forbidden Zones of Solids (DFZS) [37].

Figure 4

Comparison of $\mathrm{\Delta }{H}_f$ between PBE and HSE.

Figure 5

$\mathrm{\Delta }{H}_f$ comparisons between DFT (left: PBE, right: HSE) and experiments.

Figure 6

Comparisons of errors in calculating the formation energies via PBE and HSE relative to experiment. Compounds are partitioned into groups: all compounds, strongly bound compounds, weakly bound compounds, all metals, small-bandgap compounds, and large-bandgap compounds. ME, MAE, and RMSE refer to mean error, mean absolute error and root-mean-square error, respectively. The numbers of compounds in each group are denoted in parentheses.

Figure 7

Left: DFT errors in prediction $\mathrm{\Delta }{H}_f$ with respect to calculated bandgaps for nonzero bandgap compounds. Right: Box plots for DFT calculation errors in formation energy for nonzero bandgap compounds.

Figure 8

Mean absolute errors in DFT estimation of $\mathrm{\Delta }{H}_f$ using PBE and HSE when compared against experimental data among different classes of binary compounds.

Figure 9

Elemental chemical potential corrections $\delta \mu$ obtained via fitting for PBE and HSE. Within each cell, the upper triangle shows HSE chemical potential corrections, and the lower triangle shows PBE corrections. Elements with positive corrections are colored red, and negative ones are colored blue. The elements for which we are currently unable to compute the chemical potentials are enclosed in solid squares without a diagonal line.

Figure 10

MAE and RMSE for corrected PBE and HSE formation energies for the cases of fitting $\mu$ for no elements (“Fit-none”) and all elements (“Fit-all”).

Figure 11

Multifidelity cokriging model benchmarking results for the HSE bandgap dataset of 1135 materials. Some of the full training data are split and set aside as test data during the training of the model. The trained model's predictions on the test data are plotted. The plot in (a) has an optimistic train:test split of 8:2, which imitates a situation where training data are large enough to reliably learn the correlation between input features and the target property $E_g^{\mathrm{HSE}}$. In such a situation, the vector space spanned by input features is sufficiently sampled by the training data. The plot in (b) shows the predictions from a different model trained and tested on a pessimistic train:test data split of 1:9. The pessimistic benchmarking was done to examine the prediction capability of the cokriging model in situations where training data are not large enough to fully represent the relatively larger portion of the feature space spanned by the candidate materials. In (c), the uncertainty quantified for each test data point by the cokriging model during the prediction is analyzed.

Figure 12

(a) Bandgap openings predicted by the multifidelity model. $E_g^{\mathrm{predHSE}}$ refers to the HSE-fidelity bandgap value predicted from cokriging. In about 30% of all materials in the search space, the PBE results and cokriging model predictions disagree on whether a material is metallic or not. (b) Elemental distributions among the materials in the search space with the highest values for model uncertainty. The ordinate of the bar plot represents the percentage of the compounds with a high uncertainty prediction among all the compounds in the search space that contain the element specified on the abscissa of the plot. The bars of only those elements that have an ordinate value of more than 2% are shown. Such a cutoff is kept to make the relevant information stand out and skip other elements such as O, F, etc., which have less than 2% of the compounds predicted to have a high uncertainty value. The model uncertainty is quantified as the standard deviation of the predicted cokriging Gaussian distribution of $E_g^{\mathrm{predHSE}}$.