  The self-consistent procedure in electronic structure calculations is
revisited using a highly efficient and robust algorithm for solving the
non-linear eigenvector problem i.e. H({{\psi}}){\psi} = E{\psi}. This new
scheme is derived from a generalization of the FEAST eigenvalue algorithm to
account for the non-linearity of the Hamiltonian with the occupied
eigenvectors. Using a series of numerical examples and the DFT-Kohn/Sham model,
it will be shown that our approach can outperform the traditional SCF
mixing-scheme techniques by providing a higher converge rate, convergence to
the correct solution regardless of the choice of the initial guess, and a
significant reduction of the eigenvalue solve time in simulations.
