Most commonly, the cumulant expansion is truncated at second order, such that G 2 (t) = g 2 (t), which yields the exact lineshape within the Condon approximation for a two-level system where the energy gap fluctuations follow Gaussian statistics. For non-Gaussian energy gap fluctuations, which can be induced by anharmonic PESs and mismatches in ground and excited state PES curvatures, higher order cumulants can become important. While it is possible to compute these higher order correction terms to the lineshape function in both model potentials and realistic systems, for the purpose of this work we limit ourselves to the second order cumulant approximation only. The second order cumulant g 2 (t) has the closed-form expression