In Fig. B), we focus on the relative performance between the dampened and the undampened third order cumulant approximation. We note that in the negative skew region, the dampened third order cumulant approximation is indistinguishable from the pure third order cumulant approximation in average performance, again indicating that the pure third order cumulant approximation systematically improves the lineshape in this region. In the positive yet small skew limit, it is generally observed that physically reasonable GBOMs only have a very small third order correction, resulting in very little effect in the overall response function by conditioning the cumulant expansion through the ideal dampening factor Φ. In the larger valued positive skew region, we observe a strong and consistent improvement of the dampened third order cumulant approximation over the pure third order cumulant approximation, with Φ becoming small for large skewness values. We find these results consistent with the principles that led us to propose the functional form of Eqn. 16: As we reach regions with a more strong departure from Gaussian behavior, the size of the third order cumulant contribution must increase. In parallel, the (unaccounted for) higher order cumulant contributions must increase in size as well, requiring a larger degree of dampening. Once the function Φ(µ (3) , µ (4) ) has been parameterized by fitting a bivariate spline to the sampled data points of ≈125,000 GBOMs, the prefactor conditioning of the third order cumulant approximation can now be applied in a predictive manner. To do so, we apply the prefactor method to a set of GBOMs where, rather than selecting the ideal dampening factor from minimizing the metric κ(Φ), we evaluate Φ directly from the statistical properties of the energy gap fluctuations, namely the skew µ (3) and excess kurtosis µ (4) , without the need to refer to the exact analytical solution of the spectral lineshape. Three example GBOMs can be found in Fig. (information on model parameters is provided in Appendix A 5 b). As can be seen, the fitted spline function for Φ(µ (3) , µ (4) ) yields an appropriate dampening factor for all GBOM parameterizations in different areas of statistical space. In all cases, the dampened third order cumulant lineshape is much improved over the pure third order cumulant lineshape, removing strong divergences and unphysical negative absorbances in the case of Fig. ) and C). The dampened lineshape also provides a significant improvement over the second order cumulant approximation, both in terms of the position and intensity of vibronic peaks. While Fig. shows three representative GBOM parameterizations, we observe a similar performance over a wide range of model parameterizations studied. The fact that the correction factor can be applied with good accuracy to the same class of models it was parameterized for is not entirely surprising, but serves at a good empirical demonstration that Φ can be truly represented as a function of the non-Gaussian features of the energy gap fluctuations, rather than explicit parameters of the underlying Hamiltonian. In the context of GBOMs, this implies that the optimal prefactor within a region of non-Gaussian fluctuations obtained from a set of randomly sampled GBOMs can still be effectively applied to a GBOM that may strongly vary in underlying parameters, as long as it shares similar energy gap fluctuation statistics. What remains to be demonstrated is that a dampening factor Φ(µ (3) , µ (4) ) derived for a simple set of harmonic model can be applied to more general (anharmonic) systems. However, the good performance of the dampening factor shown in Fig. is highly promising, as it suggests that failures of the cumulant approach in correctly capturing Duschinsky mode mixing effects and changes in PES curvature upon excitation can be effectively cured even in complex condensed phase systems, where the FC method cannot be applied.