A Machine Learning-Driven Methodology for the Design of Exchange-Correlation Functionals

Abstract

Computational chemistry and materials science aim to explain and predict the behavior of chemical systems using computational models. To achieve this goal, we require a method that accurately describes the ground state electronic structure of chemical systems. Due to its combination of computational efficiency and accuracy, density functional theory (DFT) is the most popular tool for these calculations. However, while DFT is exact in principle, the formalism contains a term called the exchange-correlation (XC) functional, which provides the energy difference between the exact interacting-electron system and a model non-interacting system. This XC functional is unknown and therefore must be approximated in practice, and this approximation is the key limiting factor in the accuracy of DFT.

Recently, machine learning (ML) has gained attention as a means to develop more accurate XC functionals. While significant progress has been made in this direction, it has proven difficult to overcome the persistent trade-offs between accuracy, computational efficiency, numerical stability, and chemical transferability. To address this problem, we developed a framework called CIDER for learning XC functionals that are accurate, transferable, and efficient. CIDER consists of two key components. First, we use a Bayesian machine learning model called Gaussian process regression to learn functional forms that can be carefully tuned to balance accuracy and smoothness, an important trade-off in functional design. Second, we design nonlocal features of the density that enable the model to obey exact physical constraints on the exchange functional, and we implement computationally efficient algorithms to evaluate these features within different types of DFT software. The combination of fast, nonlocal input features with a flexible and tunable ML model enables the accurate description of molecular and solid-state systems within a single model. We also extend the CIDER framework to explicitly fit band gaps and other electronic properties, thereby addressing the infamous band gap problem of DFT within a machine learning framework.

In this dissertation, I will first provide introductions to DFT and Gaussian process regression, and then I will provide an overview of existing research on ML XC functionals and outstanding challenges in the field. After that, I will describe the CIDER framework in detail, including the theoretical justification for the structure of the XC functional, the computationally efficient implementation of the models, and the application of CIDER functionals to physically complex problems requiring large-scale materials simulations, such as point defects in semiconductors and polarons in ionic crystals.