Investigating Ground-State Electronic Structure in the Era of Machine Learning

Abstract

Understanding how electrons behave is one of the most fundamental problems in physics, chemistry, and materials science, as it underpins the properties of matter. However, solving the many-electron Schrödinger equation remains an immense challenge due to the complexity of electron-electron interactions. In this thesis, we investigate how recent advances in machine learning can provide powerful new approaches to tackle this pivotal electronic structure problem. This thesis consists of two main chapters: First, we apply neural network wavefunctions and stochastic optimization techniques within quantum Monte Carlo frameworks to study correlated electron systems. We employ neural network wavefunctions in variational Monte Carlo to study the two-dimensional electron gas, and automatically discover different phases of the system, including floating Wigner crystals and nematic spin correlated liquids. We also apply the variational principle and automatic differentiation to improve auxiliary field quantum Monte Carlo calculations for molecules, achieving significantly improved accuracy with mild computational scaling. Second, we develop data-driven exchange-correlation functionals for density functional theory (DFT) using machine learning. We develop a new energy functional representation that maps reduced one-body density matrices to energy using deep neural networks. We also propose a self-consistent optimization scheme using diverse data labels to integrate the functional with the Kohn-Sham framework. Combining these ingredients, we achieve chemical accuracy for a wide range of molecular systems at a cost similar to DFT methods, while requiring fewer training labels. Our work demonstrates the transformative potential of machine learning in solving complex quantum systems, paving the way for more efficient and accurate electronic structure modeling.