Summer School on Quantum and Kinetic Theory for Complex Systems

Density Functional Theory: models and numerical methods

Eric Cances

Ecole des Ponts and INRIA

First-principle molecular simulation based on electronic structure calculation has become an essential tool in chemistry, condensed matter physics, molecular biology, materials science, and nanosciences. In this lecture, I will focus on Density Functional Theory and the Kohn-Sham model, which is to date the most widely used approach in electronic structure calculation, since it provides the best compromise between accuracy and computational efficiency. The Kohn-Sham model is a constrained optimization problem, whose Euler-Lagrange equations have the form of a coupled system of nonlinear elliptic eigenvalue problems. I will explain how to derive the Kohn-Sham model from the N-body Schrödinger equation, comment on the approximations made, and present some numerical methods to solve this problem.