**Description:**

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.

**Contents:**

Preface

**Chapter 1: Basic Theory of
quantum mechanics** • Finite dimensional quantum
systems • Schrödinger equation in the real space • Hydrogen atom • Periodic
systems • Tensor product spaces: Twin spin -½ particles • Identical particles

**Chapter 2: Density functional
theory: Formulation and algorithms** • Hartee-Fock
theory • Kohn-Sham density functional theory • Nonlinear eigenvalue problem •
Self-consistent field iteration • Density matrix formulation • Extension to
finite temperature • Density matrix algorithms • Brillouin zone sampling for
periodic systems • Localization • Geometry optimization and ab initio molecular
dynamics • Time-dependent density functional theory

**Chapter 3: Linear response theory** • Perturbation of Green’s function • Perturbation of the density
matrix • Density functional perturbation theory • Applications of density
functional perturbation theory • Exponential decay of the Green’s function •
Time-dependent density functional perturbation theory • Casida formalism •
Random phase approximation

**A.
Notations and preliminaries **

A.1
Notation • A.2 Spherical harmonics • A.3 Equalities in complex analysis

**B.
Selected references for further reading**

**Bibliography**

**Index**

**About the Authors:**

**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.

**Target Audience:**

The
book is written for advanced undergraduate and beginning graduate students,
specifically those with mathematical backgrounds but without a prior 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. People interested in physics, chemistry,
materials science, and related fields along with applied and computational
mathematics.