Error estimates for solid-state density-functional theory predictions: an overview by means of the ground-state elemental crystals (1204.2733v5)

Abstract: Predictions of observable properties by density-functional theory calculations (DFT) are used increasingly often in experimental condensed-matter physics and materials engineering as data. These predictions are used to analyze recent measurements, or to plan future experiments. Increasingly more experimental scientists in these fields therefore face the natural question: what is the expected error for such an ab initio prediction? Information and experience about this question is scattered over two decades of literature. The present review aims to summarize and quantify this implicit knowledge. This leads to a practical protocol that allows any scientist - experimental or theoretical - to determine justifiable error estimates for many basic property predictions, without having to perform additional DFT calculations. A central role is played by a large and diverse test set of crystalline solids, containing all ground-state elemental crystals (except most lanthanides). For several properties of each crystal, the difference between DFT results and experimental values is assessed. We discuss trends in these deviations and review explanations suggested in the literature. A prerequisite for such an error analysis is that different implementations of the same first-principles formalism provide the same predictions. Therefore, the reproducibility of predictions across several mainstream methods and codes is discussed too. A quality factor Delta expresses the spread in predictions from two distinct DFT implementations by a single number. To compare the PAW method to the highly accurate APW+lo approach, a code assessment of VASP and GPAW with respect to WIEN2k yields Delta values of 1.9 and 3.3 meV/atom, respectively. These differences are an order of magnitude smaller than the typical difference with experiment, and therefore predictions by APW+lo and PAW are for practical purposes identical.

• The paper quantifies DFT prediction errors by statistically separating systematic deviations from residual numerical uncertainties in elemental crystals.
• It reveals volume overestimations of ~3.6% and bulk modulus underestimations of ~4.9% with residual error bars of 1.1 ų/atom and 15 GPa, respectively.
• The study underscores the need for improved DFT methods to better capture van der Waals interactions and enhance reproducibility across implementations.

Error Estimates for Solid-State Density-Functional Theory Predictions

The paper "Error estimates for solid-state density-functional theory predictions: an overview by means of the ground-state elemental crystals" by Lejaeghere et al. aims to provide a comprehensive evaluation of density-functional theory (DFT) in predicting properties of crystalline solids. Through a carefully curated set of elemental crystals, this paper tackles a crucial question that experimental and theoretical materials scientists face: what is the expected error in such first-principles predictions?

Summary of the Approach

The paper adopts an empirical approach by using a test set composed of all ground-state elemental crystals, with an intentional exclusion of most lanthanides. By comparing DFT predictions to experimental results across multiple properties such as cohesive energy, volume, bulk modulus, and elastic constants, the authors derive both intrinsic and numerical error estimates for DFT.

The focus is on deriving these errors from a statistical analysis, segregating systematic deviations from residual error bars. The paper also touches upon the reproducibility of predictions across popular DFT implementations like VASP and WIEN2k, using descriptors like the quality factor, denoted as $\Delta$.

Key Findings and Numerical Results

The paper reports a consistent behavior of DFT within the generalized gradient approximation (GGA), as implemented in the Perdew-Burke-Ernzerhof (PBE) functional. The intrinsic systematic errors and residual error bars are quantified for several properties:

• Volume predictions are found to be systematically overestimated by around 3.6%, a likely consequence of the typical underbinding characteristic of GGA functionals. The corresponding residual error bar is relatively tight at 1.1 ų/atom, demonstrating reliable volume predictions for most materials.
• Bulk modulus predictions tend to be underestimated by approximately 4.9%, with a 15 GPa error bar, highlighting the challenge of accurate compressibility predictions within this framework.
• The typical behavior of DFT-GGA includes an inability to effectively account for van der Waals (vdW) interactions, significantly affecting the accuracy for molecular crystals and noble gases.

Implications and Conclusion

This work has profound implications for the scientific community leveraging DFT in solid-state material predictions. Firstly, it emphasizes the need for systematic error quantification when utilizing DFT predictions for materials design and analysis. Secondly, it illustrates the necessity of considering both intrinsic and numerical errors when interpreting DFT results—particularly when experimental discrepancies arise.

The results suggest that while GGA is an effective and widely adopted approach, the inability to model certain interactions like vdW forces may call for the exploration of newer functionals or correction schemes. The systematic quantification of how various DFT implementations compare further aids researchers in selecting appropriate computational tools.

Moreover, the paper underscores the importance of ongoing development in DFT methodologies to include effects such as electron correlation, spin-orbit coupling, and London dispersion, which could potentially lead to more accurate predictions and reduced discrepancies with experiment.

Finally, by offering a protocol for estimating and correcting systematics in DFT predictions, this work serves as a practical reference for researchers aiming to boost the reliability of their computational materials predictions. The ongoing extension of error benchmarks to incorporate emerging computational methods would enhance the community's confidence in DFT as a tool for predictive materials science. Future research might benefit from extending this systematic analysis to hybrid functionals, meta-GGA, and many-body perturbation theories to explore new avenues for improving predictive accuracy.