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 talk, 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 present some recent progress made in the numerical analysis of this model, which paves the road to high-fidelity numerical simulations (with a posteriori error bounds) of the electronic structure of large molecular systems.