Harvard researchers present a Gaussian-based machine learning approach that tunes to the energy levels of individual particles

One of the main challenges of semilocal density functional theory (DFT) is the consistent underestimation of band gaps, mainly due to self-interaction and delocalization errors. This problem complicates the prediction of electronic properties and charge transfer mechanisms. Hybrid DFT, incorporating a fraction of the exact exchange energy, offers improved band gap predictions but often requires system-specific tuning. Machine learning approaches have emerged to improve DFT accuracy, in particular for molecular reaction energies and strongly correlated systems. Explicit tuning of energy gaps, as demonstrated by the DM21 functional, could improve DFT predictions by addressing self-interaction errors.

Harvard SEAS researchers have developed a machine learning method that uses Gaussian processes to improve the accuracy of density functionals for predicting energy gaps and reaction energies. Their model integrates nonlocal features of the density matrix to accurately forecast molecular energy gaps and estimate polaron formation energies in solids, despite being trained exclusively on molecular data. This advance builds on the CIDER framework, which is known for its efficiency and scalability in handling large systems. Although the model currently targets exchange energy, it holds promise for broader applications, including predicting electronic properties such as band gaps.

The study presents key concepts and methods for fitting exchange-correlation (XC) functionals in DFT, focusing on the prediction of band gaps and single-particle energies. It covers the application of Gaussian process regression to model XC functionals, incorporating training features that improve accuracy. The theoretical foundation is based on Janak’s theorem and the concept of derived discontinuity, which are crucial for predicting properties such as ionization potentials, electron affinities, and band gaps within generalized Kohn–Sham DFT. The approach aims to improve functional training by addressing the challenges of orbital occupancy and derived discontinuities.

The CIDER24X energy exchange model was developed using a Gaussian process, which improves flexibility. Key features were chosen to achieve minimum covariance, tuned to a specific range, and used to train a dense neural network approximating the Gaussian process. Training data comprised uniform electron gas exchange energy, molecular energy differences, and energy levels obtained from databases such as W4-11, G21IP, and 3d-SSIP30. Two variants of the CIDER24X model were created: one incorporating energy level data (CIDER24X-e) and one excluding it (CIDER24X-ne) to evaluate the effect of including energy levels in the fitting process.

The study presents the CIDER24X model with SDMX features, which shows improved predictive accuracy for molecular energies and HOMO-LUMO gaps compared to previous models and semilocal functionals. CIDER24X-ne, trained without energy levels, aligns closely with PBE0, while CIDER24X-e, which includes energy level data, offers a better balance between energy and band gap predictions. Although there are trade-offs, particularly with eigenvalue training, CIDER24X-e outperforms semilocal functionals and approaches the accuracy of hybrid DFT, making it a promising alternative that reduces computational costs.

In conclusion, the study presents a framework for fitting density functionals at the bulk and single-particle energy levels by machine learning, leveraging Janak's theorem. A new feature set, SDMX, is introduced to learn the exchange functional without requiring the full exchange operator. The model, CIDER24X-e, maintains accuracy in molecular energies while significantly improving energy gap predictions, matching hybrid DFT results. The framework is extensible to full XC functionals and other machine learning models, offering the potential for efficient and accurate electronic property predictions in diverse systems.

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