The metric κ is simply taken as the unsigned difference between the cumulant and the exact spectrum, averaged over n numerical grid points along the frequency axis. We evaluate σ cumul using Eqn. 16, where the parameter Φ is systematically varied from 0 (corresponding to a pure second order cumulant spectrum) to 1 (corresponding to an undampened cumulant expansion truncated at third order). A plot of the resulting κ(Φ) can be found in Fig. ). We find that κ(Φ) is a convex function, and thus by selecting the prefactor value corresponding to the minimum of κ(Φ), one may optimize the performance of the third order cumulant approximation with respect to the exact spectrum. Selecting the optimal dampening factor Φ when constructing the third order cumulant lineshape following Eqn. 16 results in a spectral lineshape in very close agreement to the exact spectrum, both in the intensity and positioning of vibronic peaks (see Fig. B and SI Sec. V). Additionally, the unphysical negative absorbance is removed from the spectrum. While Fig. shows results for a specific parameterization of the GBOM Hamiltonian, we find that similar results can be obtained over a wide parameter range, suggesting that the functional form of a dampened cumulant response proposed in Eqn. 16 is widely applicable.