the reaction rate constant will be enhanced or suppressed, typically up to 4-5 times compared to the rate constants outside the cavity. If we define |ν L as the vibrational ground state of the reactant (left well), and |ν L as the vibrationally excited state of the reactant, then ω 0 corresponds to the frequency of the |ν L → |ν L transition. An experimental review summarizing the recent advances in VSC-modified chemical reactions can be found in Ref. Electronic mail: wying3@ur.rochester.edu b) Electronic mail: pengfei.huo@rochester.edu whereas general discussions of this topic can be found in many recent reviews. This new strategy of VSC, if feasible, will allow one to bypass some intrinsic difficulties (such as intramolecular vibrational energy transfer) encountered in mode-selective chemistry that uses infrared (IR) excitation to tune chemical reactivities, offering a paradigm shift of synthetic chemistry through cavityenabled bond-selective chemical transformations , since one can use the cavity to selectively slow down one competing reaction over the target reactions. Unfortunately, a clear theoretical understanding of cavity-modified ground-state chemical reactivity remains missing, despite recent theoretical progress. Currently, there are no well-accepted mechanisms or theoretical explanations for the observed phenomena. For electronically nonadiabatic reactions, such as electron transfer reactions coupled to optical cavities, there are numerous exciting progresses which can explain the resonance effect (Eq. 1). Their relevance related to VSC experiments, which are all electronically adiabatic reactions, has yet to be clarified. For the ground state adiabatic reaction coupled to the cavity, transition state theory (TST) predicts no modification of the rate constant nor any significant cavity frequency dependence. From Grote-Hynes (GH) theory, it was conjectured that the cavity effect is purely from the modification of the transmission coefficients due to the dynamical caging effect (where the cavity mode acts as a regular nuclear vibration that provides friction to the reaction coordinate), providing a cavity frequency dependent VSC modification of the rate constants. plain the cavity frequency, ω c , dependence of VSC modifications, they often cannot predict the correct resonant frequency that matches the quantum vibration frequency ω 0 measured from the optical spectra. If one describes the rate constant using classical theory, such as the classical flux-side correlation function, the Grote-Hynes (GH) theory, the maximum modification of the rate constant occurs when ω c ≈ ω b , where ω b is the top of the barrier frequency (imaginary frequency of the Transition State). The Pollak-Grabert-Hänggi (PGH) theory, or a semi-classical version of the quantum transition state theory (q-TST) rate theory, finds that the cavity-frequency dependent rate modification is related to either the top of the barrier frequency ω b (Eq. 21), the classical bottom of the well frequency ω cl 0 (Eq. 26), or a broad frequency distribution between these two frequencies. For example, the GH theory for a double-well (DW) model in the spatial diffusion-limited regime (after the Kramers turnover) predicts that the VSC modification is sensitive to ω b . The GH theory for a solvent-solute model coupled to the cavity predicts the VSC effect is sensitive to a frequency that depends on both ω b and ω cl 0 . The q-TST theory predicts that the VSC-modified rate will have a broad profile that spans the range of ω b and ω cl 0 . The PGH theory for a DW system coupled to the cavity in the energy diffusion-limited regime (before the Kramers turnover) predicts that the rate modification could be closer to ω cl 0 (see Fig. of Ref. 23) or closer to ω b (see Fig. of Ref. 25), due to the fact that the rate profile in this energy diffusion-limited regime is dominated by the energy loss process of the classical trajectory that travels between the bottom of the well and the top of the barrier. Related to the classical rate theory, a direct abinitio simulation that investigates the reaction in Ref. 1 treats the electronic ground states ab-initially and treats nuclear and photonic degrees of freedom (DOFs) classically. The simulation suggests that the bond distance (of the reactive chemical bond) will be modified when the cavity frequency is close to either ω b or ω cl 0 (note that this is not directly related to the rate constant). Overall, in these classical/semi-classical rate theories, there is no knowledge directly related to the quantum vibrational ground and excited states and the associated transition frequency ω 0 , except for the pure harmonic potential that ω cl 0 = ω 0 . The only explicitly available frequencies are those ω cl 0 and ω b that are directly related to the potential V ( R) (Eq. 20). In this sense, the quantum vibration frequency ω 0 is intrinsically quantum information, as it is directly related to the vibrational eigenenergies (by solving the eigenequation and does not have a classical analogy when the potential energy surface is anharmonic). Interestingly, the purely classical description of the vibrational and photonic DOFs will provide different peak positions of the optical spectra and the rate modification profile. For example, as demonstrated in Ref. 25, the classical IR spectra of the molecule peaks at ω cl 0 (because the classical trajectory predominantly samples the bottom of the well region), and the VSC-modified rate profile peaks at ω b , thus having a large frequency difference. Simply adding quantum statistics and quantum tunneling using ring polymer molecular dynamics (RPMD) will not produce the correct resonance frequency and sharp resonance either because there is no explicit information of ω 0 in the ring polymer Hamiltonian. As also extensively discussed in Ref. 25, the exact quantum optical spectra and VSC-modified rate profile both peak at ω 0 , because they have a common origin related to the quantum transition of |ν L → |ν L . Our work confirms the same discovery by treating qc as a bath DOF with spectral density description.