Here, E e i and E g i denote the eigenvalues corresponding to the ith eigenstate of the nuclear Hamiltonians Ĥe and Ĥg respectively, P(i) is the Boltzmann population of the ith energy level on the ground state PES in thermal equilibrium and ω ji is the energy required to transition from state i on the ground state PES to state j on the excited state PES. For model systems where the eigenstates of the nuclear Hamiltonians can be computed exactly, the spectral decomposition yields the exact linear response function under the Condon approximation. This form provides a crucial benchmark within the scope of this work as a means to quantify the performance of approximate cumulant-based schemes (see Sec. II B). However, the sum-over-states formalism in Eqn. 5 has a number of drawbacks, especially when modeling realistic molecular systems in the condensed phase.