We have demonstrated that the parameterized dampening factor rigorously removes unphysical and divergent lineshapes in the third order cumulant approximation, both for model systems and for condensed phase systems sampled with MD. In the GBOM Hamiltonian, we have also shown that the approach improves the agreement with the exact spectrum, both compared to the second order and the pure third order cumulant approach, for a wide range of model system parameterizations. For the realistic condensed phase system of coumarin-135 in toluene, our method yields a final spectrum in excellent agreement with experiment. The results presented indicate that the approach outlined in this work provides an efficient, computationally affordable pathway for correcting the main shortcomings of the second order cumulant approach in condensed phase systems, namely the inability to account for non-Gaussian fluctuations introduced by anharmonic effects, Duschinsky mode mixing effects and changes in the PES curvature upon excitation. Since our parameterization of Φ only relies on the skewness and kurtosis of the underlying energy gap fluctuations, we expect the method to be widely applicable to condensed phase systems sampled with MD. The dampened third order cumulant approach developed in this work has been implemented in the open-source software package MolSpeckPy developed within our group. Appendix A: Model systems