For general molecular Hamiltonians, all orders of cumulants contribute to the spectral lineshape and a truncation at any finite order introduces errors. However, in the limit that the energy gap fluctuations obey Gaussian statistics, all cumulants beyond g 2 (t) vanish, such that exact response is recovered at the second order cumulant approximation. This behavior is observed for a system where all modes coupled to the electronic transition are harmonic and equal in ground and excited state curvature, resulting in an energy gap operator that is linear with respect to nuclear coordinates (See SI Sec I). Such a simplified model system is known as the Brownian oscillator model (BOM).