For all GBOMs sampled, the second and third order cumulant lineshapes were evaluated using the analytical expressions for the exact quantum correlation functions, that are generally inaccessible in realistic condensed phase systems. This was done so that the prefactor Φ(µ (3) , µ (4) ) was independent of errors introduced in the lineshape through constructing cumulants from classical correlation functions using QCFs. We have performed tests (see SI Sec. V A) to confirm that the errors introduced through QCFs are generally small in comparison to the errors introduced by a low-order truncation of the cumulant expansion. Thus the Φ(µ (3) , µ (4) ) derived for exact quantum correlation functions in the GBOM Hamiltonian is expected to perform well for condensed phase systems sampled in MD, where lineshape functions have to be evaluated from classical correlation functions using QCFs.