Electronic structure calculations have become an indispensable tool in chemistry, molecular biology, materials science, and nanotechnology. The density functional theory (DFT) of Hohenberg, Kohn and Sham is an approach for computing the ground-state density and energy of a many-electron system by solving a constrained minimization problem whose first order optimality conditions, the Kohn-Sham equations, can be written as a nonlinear eigenvalue problem. Used almost exclusively in condensed matter physics since the 1970's, DFT became popular in quantum chemistry in the 1990's due to the development of more accurate approximations. Today, DFT is the most widely used ab initio method in material simulations. DFT can be used to calculate the electronic structure, the charge density, the total energy, and the atomic forces of a material system; and with the advance of new algorithms and supercomputers, DFT can now be used to study thousand-atom systems. There are many challenges remaining though, especially for large systems (more than 100,000 atoms), problems requiring many total energy calculation steps (molecular dynamics or atomic relaxations), or systems with open-shell character. More accurate and better-justified approximations to the density functional for the exchange-correlation energy are also continually being developed, requiring new exact constraints and presenting new computational challenges.
Wave function methods have also known spectacular development in recent years. These methods allow, in principle, the construction of increasingly refined approximations to the many-electron Schrödinger equation. They outperform conventional DFT with respect to accuracy, but at the price of a dramatic increase in computational cost. Reducing the computational cost of wave function methods, while preserving its accuracy is one of the major challenges in quantum chemistry. Important steps in this direction have been taken with the introduction of linear scaling algorithms. Other important challenges include systems with electronic degeneracies and calculations of a wider range of properties and experimental observables.
This tutorial will focus on presenting some of the fundamental concepts and techniques currently used in electronic structure calculations. The first day will introduce some of the key ideas of quantum mechanics and wave function methods, including coupled cluster methods and DFT. This will be followed on the second day by an introduction to some of the major mathematical techniques used in the formulation and solution of electronic structure problems. We will also discuss some commonly used computational methods for solving these problems. Throughout, we will present some of the mathematical and computational challenges in developing accurate, efficient, and robust algorithms for electronic structure calculations of large systems.