Efficient Implementation of Machine Learning-Based Nonlocal Functionals for Molecules and Solids

Machine Learning (ML) has recently gained attention as a means to develop more accurate exchange-correlation (XC) functionals, and in particular could be used to improve XC functionals for solids without using the computationally expensive exact exchange energy. Feature design is one of the main challenges of this approach because the features must include enough nonlocality to capture the complex nonlocal nature of the exact XC functional while also allowing for computationally efficient and scalable implementations. To address this challenge, the CIDER model for designing nonlocal ML functionals is introduced and used to create an exchange functional that obeys the uniform scaling rule for exchange. In addition, two efficient methods for evaluating CIDER features are implemented: A quadratic-scaling algorithm for all-electron, molecular density functional theory (DFT) and a quasi-linear-scaling algorithm for plane-wave DFT with the PAW method. Efficiency and accuracy benchmarks for molecular periodic systems are presented, and the importance of the kinetic energy as a feature for learning the exchange functional is discussed.