Addressing the Band Gap Problem with a Machine-Learned Exchange Functional

Abstract

The systematic underestimation of band gaps is one of the most fundamental challenges in semilocal density functional theory (DFT). In addition to hindering the application of DFT to predicting electronic properties, the band gap problem is intimately related to self-interaction and delocalization errors, which make the study of charge transfer mechanisms with DFT difficult. In this work, we present two key innovations to address the band gap problem. First, we design an approach for machine learning density functionals based on Gaussian processes to explicitly fit single-particle energy levels. Second, we introduce novel nonlocal features of the density matrix that are expressive enough to fit these single-particle levels. Combining these developments, we train a machine-learned functional for the exact exchange energy that predicts molecular energy gaps and reaction energies of a wide range of molecules in excellent agreement with reference hybrid DFT calculations. In addition, while being trained solely on molecular data, our model predicts reasonable formation energies of polarons in solids, showcasing its transferability and robustness. Our approach generalizes straightforwardly to full exchange-correlation functionals, thus paving the way to the design of novel state-of-the-art functionals for the prediction of electronic properties of molecules and materials.

35 pages, 5 figures, 3 tables, supporting information; v2: fixed incorrect figure 1 plot and fixed equations 58, 73, 77, and 78; v3: expanded analysis of polaron formation energies

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• Koopmans' condition for density-functional theory
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Cited by in corpus (3)

• Algorithmic differentiation for plane-wave DFT: materials design, error control and learning model parameters
• Exchange-Correlation Functionals in 2D Materials: Applications, Challenges, and Limitations
• Neural network distillation of orbital dependent density functional theory