To construct a plot of Φ µ (3) , µ (4) as outlined in Sec. II D, ≈ 125, 000 individual GBOM parameterizations (see Appendix A 1) were sampled. The number of modes in the GBOMs was varied systematically from 2 to 50. The ground and excited state frequencies and shift vectors were sampled from uniform distributions over a physically realistic range of molecular vibrations. The n-mode Duschinsky rotation matrix relating ground-to excited state normal modes was constructed by filling the off-diagonal elements of each row with randomly generated values such that their sum equals a predetermined value; this value being sampled from a uniform distribution as well. The coupling matrix was subsequently made unitary through a Gram-Schmidt orthonormalization of column vectors. Full details of the distributions of model parameters and how they relate to realistic molecular systems can be found in SI Sec. VI A.