* To be presented at the mini-symposium on "Eigenvalue Problems'' organized by H. van der Vorst.

Physical properties of matter can be determined by solving a coupled system involving Schr\"odinger's equation coupled with Poisson's equation. This coupling is nonlinear and rather complex. It involves a charge density $\rho$ which can be computed from the wavefunctions $\psi_i$, for all occupied states. However, the wavefunctions $\psi_i$ are the solution of the eigenvalue problem resulting from Schr\"odinger's equation whose coefficients depend nonlinearly on the charge density. This gives rise to a non-linear eigenvalue problem which is solved by a so-called Self Consistent Field (SCF) iteration. The challenge comes from the large number of eigenfunctions to be computed for realistic systems which consist of hundreds or thousands of atoms. We will discuss a parallel implementation of a finite difference approach for this problem and report on some results. We will also explore the fundamental underlying linear algebra which can be viewed as a problem of determining the diagonal of a projector associated with an invariant subspace. Methods that avoid completely the computation of eigenvectors will be briefly outlined.