In contrast to the skewed Gaussian profile of the overdamped spectral density, the underdamped spectral density features a sharp Lorentzian peak at the intramolecular mode frequency with width determined by the damping parameter γ 1 . For this approach an underdamped HEOM is derived from the multi-component spectral density resulting in a multi-component HEOM. Note that when the vibration is subsumed into the bath, γ 1 introduces additional damping which is not present when the vibration is contained in the Hamiltonian. Approaching the limit γ 1 → 0, the two approaches become equivalent as discussed within ref. , but in practice some additional damping from the second bath is unavoidable.