For the fixed energy level between |ν L ⟩ and |ν ′ L ⟩ states, the FGR rate constant is expressed as k VSC = κ(ω 0 ), where ℏω 0 is the energy difference for the transition. However, R 0 also couples to its own solvent bath (see Eq. 29), and has fluctuating energy levels. To account for this effect, we add a broadening function A 0 (ω -ω 0 ) to the frequency ω 0 , with a Lorentzian shape (for the case under the homogeneous limit ) expressed as