In Quantum Chemistry, an important class of models involve a system of coupled nonlinear eigenvalue problems which are the Euler-Lagrange equations of an energy functional. From a mathematical viewpoint, variational methods are the most powerful tool for proving the existence of a solution, which is found as a critical point (often a minimizer) of the energy. In practice, however, a fixed-point algorithm commonly called Roothaan iteration is widely used in the Quantum Chemistry community, to find numerical solutions. This algorithm does not fully exploit the variational structure of the problem, and it does not always converge. The aim of this talk is to describe (old and new) alternative algorithms combining variational and fixed-point ideas, and to give rigorous convergence results. Part of this talk is based on joint work with Guillaume Legendre and Mi Song Dupuy.