Méthodes mathématiques en chimie quantique : une introduction / Eric Cancès, Claude Le Bris, Yvon Maday

This book presents the mathematical foundations of several models of quantum chemistry. It is intended for graduate students in mathematics (and possibly also for scientists from physics or chemistry interested in understanding the formal underpinnings of their models), and introduces techniques and methods from several mathematical fields, in particular variational techniques, nonlinear analysis, spectral theory and partial differential equations theory. Chapter 1 gives an overview of some classical approximations of the electronic ground state problem, which consists in finding the lowest eigenvalue of the Schrödinger operator. The various models presented are then studied in the following chapters of the book. Chapters 2, 3 and 4 are concerned with variational problems in the Thomas-Fermi vein, on bounded or unbounded domains, and possibly also for non-convex functionals (for this last case, the concentration-compactness principle is mentioned). Chapters 6, 7 and 8 are devoted to the Hartree-Fock problem, which is a nonlinear eigenvalue problem. Since this model is of great practical importance (in itself, or as a building block for more advanced methods), two chapters describe the numerical analysis of the standard fixed-point strategy used to solve it. Chapters 9 and 10 present results for models of condensed matter, while Chapter 11 delineates some extensions, either from the mathematical viewpoint, or from the applications' side. The books ends with two appendices, the first one introducing the concepts of quantum mechanics in a way suitable for a mathematician (highlighting the assumptions, axioms, mathematical structure of the theory), the second one presenting the basics of the spectral theory of unbounded self-adjoint operators. (MathSciNet)