The possibility of calculating vibrationally resolved spectra from very short numerically exact and approximate quantum dynamical propagations using the new Filter-Diagonalization method is explored. The benchmark process under study concerns electron photodetachment in the I^-Ar₂ complex. Comparison with results obtained from long time propagations and with experiment reveals the power of the Filter-Diagonalization scheme. Using the new methodology it now becomes possible to extract positions of spectral peaks for large polyatomic systems from approximate quantum propagations e.g., by means of the recently developed Classical Separable Potential approach.