A Mathematical Introduction to Electronic Structure Theory

Based on first principle quantum mechanics, electronic structure theory is widely used in physics, chemistry, materials science, and related fields and has recently received increasing research attention in applied and computational mathematics. This book provides a self-contained, mathematically oriented introduction to the subject and its associated algorithms and analysis. It will help applied mathematics students and researchers with minimal background in physics understand the basics of electronic structure theory and prepare them to conduct research in this area. A Mathematical Introduction to Electronic Structure Theory begins with an elementary introduction of quantum mechanics, including the uncertainty principle and the Hartree Fock theory, which is considered the starting point of modern electronic structure theory. The authors then provide an in-depth discussion of two carefully selected topics that are directly related to several aspects of modern electronic structure calculations: density matrix based algorithms and linear response theory. Chapter 2 introduces the Kohn Sham density functional theory with a focus on the density matrix based numerical algorithms, and Chapter 3 introduces linear response theory, which provides a unified viewpoint of several important phenomena in physics and numerics. An understanding of these topics will prepare readers for more advanced topics in this field. The book concludes with the random phase approximation to the correlation energy. The book is written for advanced undergraduate and beginning graduate students, specifically those with mathematical backgrounds but without a priori knowledge of quantum mechanics, and can be used for self-study by researchers, instructors, and other scientists. The book can also serve as a starting point to learn about many-body perturbation theory, a topic at the frontier of the study of interacting electrons.

Lin Lin is an associate professor in the department of mathematics at the University of California, Berkeley, and is a faculty scientist at Lawrence Berkeley National Laboratory. He is a recipient of the Sloan Fellowship, the National Science Foundation CAREER Award, the Department of Energy Early Career Award, and the SIAM Computational Science and Engineering (CSE) Early Career Award. His research focuses on the development of efficient numerical methods for electronic structure calculations. Jianfeng Lu is an associate professor of mathematics, physics, and chemistry at Duke University, where he works in mathematical analysis and algorithm development for problems and challenges arising from computational physics, theoretical chemistry, and materials science. His work has been recognized by a Sloan Fellowship, a National Science Foundation Career Award, and the IMA Prize in Mathematics and its Applications.