where quency is ω 0 = (E ′ -E)/h which corresponds to the frequency of the |ν L ⟩ → |ν ′ L ⟩ transition. This subspace is also used to evaluate the coupling operators between the quantum DOF R0 and the classical DOF {x i , Xj }, includingi c i xi • P R0 P (from Ĥν in Eq. 3) andj c j Xj • P R0 P (from Ĥeff in Eq. 5), with the detailed expressions in Supporting Information, Sec. III. Note that these quantum-classical couplings include both diagonal terms (Holstein coupling) and off-diagonal terms (Peierls couplings).