Density functional theory (DFT) makes the Schrodinger equation tractable by modelling the electronic correlation as a function of density. Its relatively modest $O(N^3)$ scaling makes it the standard method in electronic structure computation for condensed phases containing up to thousands of atoms. Computationally, its bottleneck is the partial diagonalization of a Hamiltonian operator, which is usually not formed explicitly.

Using the example of the Abinit code, I will discuss the challenges involved in scaling plane-wave DFT computations to petascale supercomputers, and show how the implementation of a new method based on Chebyshev filtering results in good parallel behaviour up to tens of thousands of processors. I will also discuss some open problems in the numerical analysis of eigensolvers and extrapolation methods used to accelerate the convergence of fixed point iterations.