Instabilities and inaccuracies of multi-configuration time-dependent Hartree-Fock

We demonstrate that the widely used multi-configuration time-dependent Hartree-Fock method is restricted to a certain class of applications and fails for scenarios where periods of low entanglement occur during the propagation. By using illustrative and physically relevant examples, based on the Hubbard model of solid state physics, we show the existence of serious instabilities in the method itself and demonstrate that the method does not converge with respect to electron correlations. Possible cures of the approach are discussed.