  The ultrafast hole dynamics triggered by the photoexcitation of molecular
targets is a highly correlated process even for those systems, like organic
molecules, having a weakly correlated ground state. We here provide a unifying
framework and a numerically efficient matrix formulation of state-of-the-art
non-equilibrium Green's function (NEGF) methods like second-Born as well as
$GW$ and $T$-matrix without and {\em with} exchange diagrams. Numerical
simulations are presented for a paradigmatic, exactly solvable molecular system
and the shortcomings of the established NEGF methods are highlighted. We then
develop a NEGF scheme based on the Faddeev treatment of three-particle
correlations; the exceptional improvement over established methods is explained
and demonstrated. The Faddeev NEGF scheme scales linearly with the maximum
propagation time, thereby opening prospects for femtosecond simulations of
large molecules.
