Resonance theory of vibrational strong coupling enhanced polariton chemistry and the role of photonic mode lifetime

Recent experiments demonstrate polaritons under the vibrational strong coupling (VSC) regime can modify chemical reactivity. Here, we present a complete theory of VSC-modi ﬁ ed rate constants when coupling a single molecule to an optical cavity, where the role of photonic mode lifetime is understood. The analytic expression exhibits a sharp resonance behavior, where the maximum rate constant is reached when the cavity frequency matches the vibration frequency. The theory explains whyVSC rate constant modi ﬁ cation closely resembles the optical spectra of the vibration outside the cavity. Further, we discussed the temperature dependence of the VSC-modi ﬁ ed rate constants. The analytic theory agrees well with the numerically exact hierarchical equations of motion (HEOM) simulations for all explored regimes. Finally, we discussed the resonance condition at the normal incidence when considering in-plane momentum inside a Fabry-Pérot cavity.


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Wenxiang Ying 1 & Pengfei Huo 1,2 Recent experiments demonstrate polaritons under the vibrational strong coupling (VSC) regime can modify chemical reactivity.Here, we present a complete theory of VSC-modified rate constants when coupling a single molecule to an optical cavity, where the role of photonic mode lifetime is understood.The analytic expression exhibits a sharp resonance behavior, where the maximum rate constant is reached when the cavity frequency matches the vibration frequency.The theory explains why VSC rate constant modification closely resembles the optical spectra of the vibration outside the cavity.Further, we discussed the temperature dependence of the VSC-modified rate constants.The analytic theory agrees well with the numerically exact hierarchical equations of motion (HEOM) simulations for all explored regimes.Finally, we discussed the resonance condition at the normal incidence when considering in-plane momentum inside a Fabry-Pérot cavity.
Despite the encouraging progress in VSC experiments, we do want to point out that there are experimental efforts that try to reproduce the published VSC results but cannot find any apparent VSC modifications.One of them is an early attempt in ref. 18 to produce the enhancement of the hydrolysis reaction 13 but not being successful.The second one was an attempt 19 to reproduce the VSC enhancement on a hydrolysis reaction 11 coupled inside an FP cavity, but cannot reproduce the effect.On the other hand, there is a preliminary attempt to reproduce the same hydrolysis reaction coupled to a plasmonic cavity and did find an enhancement of the rate constant 20 .In that same work 20 , the authors also tried to conduct this reaction inside the FP cavity and claimed to reproduce the VSC enhancement effect.We emphasize that ref. 20 has not been published but we trust readers' own judgment on evaluating it.Overall, the reproducibility of these observed VSC effects remains an open experimental challenge and needs to be addressed in the future.Nevertheless, a good overview of technical concerns with the VSC flow cell experiment is given in ref. 21.In a different direction, recent experimental investigations 22,23 on CN radicalhydrogen atom abstraction reaction do not reveal any noticeable change in the rate constant, even though the molecular system is under the strong coupling condition.However, these seemingly null results on the VSC effect have the potential to indirectly inform the fundamental mechanism and limitations of the VSC-induced rate constant modifications, and provide insights into when VSC will not be able to change rate constants.
From the theoretical side, a clear mechanistic understanding of VSCmodified ground-state chemical reactivity remains elusive, despite the recent theoretical developments [24][25][26][27][28][29][30][31] .In particular, there is no well-accepted mechanism or analytic rate theory 31 .There are many previous attempts to apply the existing rate theories (such as transition state theory (TST), Grote-Hynes theory 27,32 , quantum TST 33 , Pollak-Grabert-Hänggi theory 28,30 , and molecular dynamics simulations 34 , etc.), with the conceptual hypothesis that the cavity mode can be viewed and treated as regular nuclear vibrations 27 .However, none of them have successfully predicted the correct resonance condition or the sharp resonance peak of the rate constant distribution 27,28,33,34 .The fact that the VSC-influenced dynamics is sensitive to the quantum frequency ω 0 also explains why the GH theory 27,32 , the PGH theory 28 , or q-TST rate theory 33 cannot correctly predict the resonance condition because these theories are often based on a partition function expression that effectively sums over all possible vibrational frequencies, and does not explicitly contain the information of ω 0 , or they are more sensitive to the curvature of the potential which is not directly related to the quantum optical frequency.This suggests that the analytic rate theory of VSC Chemistry if that exists, might have a completely new analytic form 31 that one has not encountered before in the theoretical chemistry literature.
Recent theoretical studies using a full quantum description of the vibrational degrees of freedom (DOF) and photonic DOF have successfully captured the resonance behavior under the single-molecule strong light-matter coupling regime 35,36 .We have used quantum dynamics simulations to reveal how cavity modes enhance the ground state reaction rate constant 36,37 .Specifically, we considered a double well potential coupled to a dissipative phonon bath 35,36 as a generic model for chemical reaction, depicted in Fig. 1a.A simplified mechanism for the barrier crossing is described as follows where k 1 is the rate constant for the vibrational excitation of the reactant (left well), k 2 is the rate constant of transition between the vibrational excited states of the left and right well, and k 3 corresponding to the vibrational relaxation process in the right well.Through exact quantum dynamics simulation, we observed that 36 k 1 ≪ k 2 , k 3 , making |ν L ! |ν 0 L ratelimiting.Further, we found that the role of the cavity mode qc is to promote vibrational excitation and enhance k 1 .Using the steady-state approximation and Fermi's Golden Rule (FGR) rate theory, the overall rate constant is approximated as k ≈ k 1 = k 0 + k VSC , where k 0 is the outside cavity rate constant (which means in the absence of cavity modes throughout this paper), and k VSC is the cavity-enhanced rate constant.Including the cavity mode and its loss environment in an effective spectral density 36 , k VSC can be evaluated using FGR, expressed as where τ c is the cavity lifetime, Ω R is the Rabi splitting (for a single molecule coupled to the cavity, see Eq. ( 8)), ω 0 is the vibrational frequency, and is the Bose-Einstein distribution function, where β ≡ 1/(k B T) is the inverse of temperature T, k B is the Boltzmann constant.In typical VSC experiments 1,10 , ω 0 ≈ 1200 cm −1 and room temperature 1/ β = k B T ≈ 200 cm −1 , such that βℏω 0 ≫ 1 and n(ω) can be approximated as Boltzmann distribution.Under the lossy regime (τ c ≪ Ω À1 R ), Eq. ( 2) agrees well with the numerically exact HEOM results, and has a sharp peak at However, k VSC in Eq. ( 2) breaks down when τ c ≫ Ω À1 R (the lossless regime) as it disagrees with the HEOM results (see Fig. 5 in ref. 36).This suggests that there will be a different mechanism for the VSC-modified rate constant under the lossless regime.
In this work, we present a complete mechanistic picture to understand a single molecule strongly coupled to a cavity and how VSC enhances the rate constant.In particular, we investigate how cavity lifetime τ c influences the rate constants and derive a new analytic expression of the VSC rate constant under the lossless regime, based on a mechanistic observation that the rate-limiting step is the photonic excitation and the subsequent excitation transfer between photonic and vibrational DOFs.The resulting analytic rate theory, denoted as kVSC (see Eq. ( 17)), successfully described the VSC rate constant in the lossless regime and is in excellent agreement with the numerically exact results.Not only it predicts the correct resonance behavior at ω c = ω 0 , but also gives a clear explanation for the intimate connection between the VSC-modified rate constant and the optical lineshape A ν ðω À ω 0 Þ (Eq.( 14)).To the best of our knowledge, this is the first analytic theory that explains the close connection between rate constant changes and lineshape of the vibrations.
Under the resonance condition (Eq.( 4)), kVSC is proportional to τ À1 c in the lossless regime (τ c ≫ Ω À1 R ), whereas k VSC (Eq.( 2)) is proportional to τ c in the lossy regime (τ c ≪ Ω À1 R ).Moreover, we proposed an interpolated rate expression between k VSC and kVSC to describe the crossover phenomenon for intermediate τ c , and predicted that the maximal enhancement will be reached at τ c ¼ Ω À1 R .These analytic expressions provide a complete description for the τ c turnover behavior of the VSC rate constant.Particularly, it provides a novel understanding of the physical role of cavity lifetime in VSC-modified chemical dynamics, that τ À1 c can be viewed as a friction parameter based on the Kramers theory 38,39 .Under the low friction regime (τ À1 c ≪ Ω R ), the reaction rate is limited by photonic excitation (which resembles energy diffusion) and kVSC / τ À1 c , while under the high friction regime (τ À1 c ≫ Ω R ), the reaction rate is limited by light-matter conversion (which resembles spatial diffusion), and k VSC / 1=ðτ À1 c Þ. Further, we discussed the temperature dependence of the VSC-modified rate constants and derived expressions of the effective change in activation enthalpy and entropy 4 , which also agree well with the numerical exact simulations.; |ν L ; 1 ; |ν 0 L ; 0 g (as well as the corresponding states for the right well), the cavity-loss environment promotes the photonic excitation |ν L ; 0 !|ν L ; 1 , and then the photonic excitation is converted into vibrational excitation through |ν L ; 1 !|ν 0 L ; 0 , being an additional channel provided by coupling to the cavity.The phonon bath still enables the mechanism under panel (a).
Finally, we discussed the resonance condition at the normal incidence for a Fabry-Pérot (FP) cavity with one or two-dimensional in-plane momenta 31 .

Theoretical model
The molecule-cavity Hamiltonian is expressed as where ĤM is the molecular Hamiltonian, Ĥν describes the phonon coupling to the molecular reaction coordinate, ĤLM describes the light-matter coupling (cavity-molecule interactions), and Ĥc describes the cavity loss bath.
In particular, ĤM ¼ P2 2M þ Vð RÞ, where M is the effective mass of the nuclear vibration, Vð RÞ is the ground electronic state potential energy surface modeled as a double-well potential (see "Methods", Eq. (37) for details), and R is the reaction coordinate.The light-matter interaction term is expressed as 16,35,36,40 ðâ y À âÞ are the photon mode coordinate and momentum operators, respectively, where ây and â are the creation and annihilation operators for a cavity mode, and ω c is the cavity mode frequency.Further, the single molecule, single mode light-matter coupling strength is 35,36 where ϵ 0 is the permittivity inside the cavity, and V is the effective quantization volume of that mode.In Eq. ( 6), we had explicitly assumed that the ground state dipole moment μð RÞ is linear and always aligned with the cavity polarization direction 27,35 , such that μð RÞ ¼ R. Based on the two diabatic states |ν L and |ν 0 L in the left well (see Eqs. (38) and (39) in "Method"), we define the quantum vibration frequency of the reactant as ω 0 E 0 À E ¼ 1172:2 cm À1 , which is directly related to the quantum transition of |ν L ! |ν 0 L and can be determined by spectroscopy measurements (IR or transmission spectra).The Rabi splitting from the spectral measurements is related to the light-matter coupling strength as follows where the transition dipole matrix element is defined as μ LL 0 ¼ hν L j Rjν0 L i.
See "Methods", Eqs. ( 41) and ( 42) for a detailed description of the other terms in the VSC Hamiltonian.In this work, we will use η c in Eq. ( 7) and Ω R in Eq. ( 8) as interchangeable phrases.
A schematic illustration of the model system is provided in the top panel of Fig. 1. Figure 1a-b presents the potential V(R) for the ground state along the reaction coordinate R, as well as key quantum states associated with the two different mechanisms of the VSC-modified kinetics.Specifically, Fig. 1a shows the four diabatic matter states |ν L (blue), |ν R (orange), |ν 0 L (red), |ν 0 R (green), in which the cavity is included in the bath and described by an effective spectral density J eff (ω) (see Supplementary Note 2, Section A).The major VSC enhanced reaction channel is shown in Eq. ( 1), in which k 1 is the rate-limiting step.This mechanism is confirmed for the lossy regime using the exact quantum dynamics simulations in our previous work 36,37 .Figure 1b shows several key photon-dressed vibration states.These states include |ν L ; 0 (blue), |ν R ; 0 (orange), |ν L ; 1 (magenta), |ν R ; 1 (green-yellow), |ν 0 L ; 0 (red), |ν 0 R ; 0 (green) for both the reaction coordinate R and the cavity mode qc , in which the cavity is included in the system and coupled to the photon-loss environment characterized by the spectral density J c (ω) (see Supplementary Note 2, Section B).The VSC-enhanced reaction channel is shown later in Eq. ( 11), in which the photonic excitation |ν L ; 0 !|ν L ; 1 and the conversion to vibrational excitation |ν L ; 1 !|ν 0 L ; 0 are sequential steps which together act as the rate-limiting steps.Later, we will see that the FGR rate theory constructed using Eq. ( 1) works for the lossy regime while using Eq. ( 11) works for the lossless regime.

FGR rate theory in the lossy regime
For the lossy regime (τ À1 c ≫ Ω R ), the VSC modified rate constant is expressed in Eq. ( 2) based on our recent work 36 , which sharply peaks at ω c = ω 0 .Under the resonance condition (ω c = ω 0 ), Eq. ( 2) reduces to suggesting that a larger enhancement of the rate constant will be reached with a longer τ c .Eq. ( 2) provides an excellent agreement with HEOM under this lossy regime, as is verified in the previous work 36 .When τ c further  10) (red solid line), FGR rates using kVSC in Eq. ( 17) (blue solid line), and FGR rates using k int VSC in Eq. ( 20) (gold dashed line) are presented.c Resonance peaks of k/k 0 for VSC effect under the lossless regime (τ c ≫ Ω À1 R ).The FGR rates using kVSC in Eq. ( 17) (thick solid lines) are compared to the HEOM results (open circles with thin guiding lines) under a variety of τ c values.increases, Eq. ( 2) needs to include phonon broadening effect 36 to avoid divergence when τ c → ∞, resulting in which is a convolution between Eq. ( 2) and the broadening function A ν ðω À ω 0 Þ (see Eq. ( 14)), and the fundamental scaling suggested in Eq. ( 9) is preserved.Figure 2a presents the results of k/k 0 using both the numerically exact HEOM simulations (open circles with thin guiding lines) and the analytic FGR rate theory (thick solid curves), with the light-matter coupling strength η c = 0.05 a.u.For the analytic FGR rate theory, we present the results k/k 0 = 1 + 0.5k VSC /k 0 , where k VSC is evaluated using Eq. ( 10) and k 0 is directly obtained from HEOM simulations, and an empirical re-scaling factor 0.5 is applied (see "Method", rate constant calculations).One can see that Eq. ( 10) provides an excellent agreement with the HEOM results when τ c < 100 fs.Both the resonance peak position and the width of the rate constant modifications are well captured.
Figure 2b presents the τ c -dependence of k/k 0 under the resonance condition (ω c = ω 0 ), with η c = 0.05 a.u., corresponding to a Rabi splitting of Ω R ≈ 25.09 cm −1 (based on Eq. ( 8)) or equivalently, the time scale of Rabi oscillation Ω À1 R ≈211:6 fs.The numerically exact HEOM results (blue open circles) show a turnover behavior on k/k 0 when increasing τ c from the lossy limit to the lossless limit.One can observe that the FGR curve using k VSC (Eq.( 10), red) agrees well with the left-hand side of the HEOM turnover curve, corresponding to the lossy regime where τ c < 100 fs.This is because when the cavity is lossy (with a small τ c ), the cavity mode thermalizes fast with the photon-loss bath, and τ c serves as a broadening parameter in the effective spectral density 36 .The fundamental mechanism of the rate constant enhancement is the vibrational excitation |ν L ! |ν 0 L under the influence of the effective bath (see schematic in Fig. 1a).
However, Eq. ( 9) cannot described the VSC kinetics when further increasing τ c so that the lossy regime τ c ≪ Ω À1 R is no longer satisfied.This is because as τ c increases, the photon-loss bath Ĥc no longer plays the simple role of (homogeneous) broadening, breaking the fundamental mechanistic assumption in Eq. (1).A new analytic theory for this lossless regime is needed.

FGR rate theory in the lossless regime
When the cavity is under the lossless regime (τ c ≫ Ω À1 R ), the rate-limiting step of the reaction becomes the photonic excitation |0i !|1i and the subsequent excitation energy transfer (see Fig. 1b).The VSC enhancement thus originates from the enhancement of the photonic excitation caused by the photon-loss bath Ĥc , as proposed in Ref. 35.Under this regime, the numerically exact HEOM simulations suggest the following reaction mechanism (schematically depicted in Fig. 1b) and k1 ; k2 ≪ k3 ; k4 .Note that the phonon bath Ĥν can still promote the transition |ν L ! |ν 0 L , and Eq. ( 1) is still one of the main mechanism for the reaction, either outside or inside the cavity.
According to FGR (with the system-bath partition described in Supplementary Note 2, Section B), the photonic excitation |ν L ; 0 !|ν L ; 1 rate constant k1 can be evaluated using FGR, resulting in where n(ω) is the Bose-Einstein distribution in Eq. ( 3).Details of the derivation are provided in Supplementary Note 5, section A. Note that there is no resonance behavior in k1 , and it becomes unbounded when τ c → 0. The resonance behavior and boundedness of the rate constant will be ensured by k2 associated with the |ν L ; 1 !|ν 0 L ; 0 transition, which can be evaluated as Details of the derivation are provided in Supplementary Note 5, Section B. Due to the molecular phonon bath Ĥν , one needs to further consider the broadening effect in the vibration frequency ω 0 , described by a lineshape function A ν ðω c À ω 0 Þ.Under the homogeneous limit, A ν ðω c À ω 0 Þ has a Lorentzian form as follows 41 with the broadening parameter 36,42 14) is also an approximate IR spectra function under the homogeneous broadening limit (see ref. 41, Eq. (6.67)), with the width Γ ν .The parameters used in this study give Γ ν ≈ 30.83 cm −1 , which is in line with the typical values of the molecular systems investigated in the recent VSC experiments 1,10 .As such, the rate constant k2 can be evaluated as convolution between κ2 (Eq.( 13)) and A ν ðω À ω 0 Þ (Eq.( 14)) as Further, the population dynamics from HEOM (see Supplementary Fig. 2) suggests that k1 and k2 steps can be regarded as sequential kinetic steps, and the populations of |ν L ; 1 and |ν 0 L ; 0 both reach to a steady state behavior (plateau in time).Making the steady-state approximation for the mediating state, the overall rate constant for the |ν L ; 0 which contains both the resonance structure (due to the line shape function A ν ðω c À ω 0 Þ) and the boundedness with respect to τ c .Because that |ν L ; 0 !|ν L ; 1 !|ν 0 L ; 0 is rate-limiting for the entire reaction process in Eq. ( 11), the VSC-modified rate constant can also be evaluated as k ¼ k 0 þ kVSC under the lossless regime (τ À1 c ≪ Ω R ), being valid under the FGR limit 43,44 .Similar to the lossy regime, we report where k 0 is the outside cavity rate constant, and α is an ad hoc rescaling factor due to the inaccuracy of the FGR level of theory.Practically, we found α = 0.5 will match the numerically exact results from HEOM 45 .Eq. ( 17) is the first main theoretical result in this work.This analytic theory kVSC (as well as in Eq. ( 10) for k VSC ) implies that the optical lineshape of the molecule described by A ν ðω À ω 0 Þ is intimately connected to the VSC kinetics modifications, due to the fact that both are sensitive to the vibrational quantum transition.The current theory in Eq. (17) provides an analytic answer to the early numerical observations 35,36 from HEOM simulations.Under the resonance condition (ω c = ω 0 ), Eq. ( 17) becomes which implies kVSC increases as τ c decreases, being opposed to Eq. ( 9) (under the lossy regime).When the cavity approaches the lossless regime (τ c → ∞), kVSC !0 so that there will be no cavity modifications.
One can observe in Fig. 2b that the kVSC curve (Eq.( 19), blue) agrees well with the right-hand side of the HEOM turnover curve, corresponding to the lossless regime where τ c > 500fs, although a re-scaling factor of 0.5 is multiplied to kVSC .The τ c → ∞ limit has been numerically investigated in Ref. 35, suggesting that k/k 0 increases as τ c decreases.The τ c → 0 limit has been numerically checked in ref. 36, suggesting that k/k 0 increases as τ c increases.Combining the knowledge of Eq. ( 9) and Eq. ( 19), we can predict that there will be a turnover behavior for the VSC-modified rate constant.Equivalently speaking, when τ À1 c ! 0 (small friction limit), kVSC / τ À1 c , and when τ À1 c ! 1 (large friction limit) k VSC ∝ τ c .The scaling of the VSC rate constant as a function of τ À1 c coincides with the well-known Kramers turnover 38,39 .As such, one can regard τ À1 c as the friction parameter for the photon-loss environment.A similar crossover phenomenon has also been discovered in spin relaxation kinetics in semiconductors, e.g., the D'yakonov-Perel' mechanism under different momentum scattering rates [46][47][48][49][50][51] .
Figure 2c presents the ω c -dependence of k/k 0 from the numerically exact HEOM results (open circles with thin guiding lines), and the FGR rate constant using kVSC in Eq. ( 17) (thick solid lines).One can see that kVSC agrees well with the exact results for τ c > 500 fs, and the resonance peak positions are well captured by FGR (with a re-scaling factor of 0.5 applied).In addition, the widths given by kVSC are in agreement with the HEOM results for a wide range of τ c .We also note that the long tails towards lower cavity frequencies in the FGR results disagree with the HEOM results, due to the Lorentzian lineshape function decaying slowly while n(ω c ) increases fast when decreasing ω c .

VSC-modified rate constant and the optical lineshape
Apart from predicting the correct resonance condition (ω c = ω 0 ), kVSC in Eq. ( 17) also predicts that the width of the rate constant profile is determined by the lineshape function of the molecular vibration spectra A ν ðω c À ω 0 Þ, with width Γ ν (see Eq. ( 15)).Note that kVSC is slightly broader than A ν ðω À ω 0 Þ due to the Bose-Einstein distribution function n(ω c ) (see Eq. ( 3)). Figure 3a presents k/k 0 obtained by HEOM simulations (light blue open circle and shaded area) and FGR from Eq. ( 17) (dark blue solid line), respectively, as well as the IR spectra of the bare molecule system obtained from HEOM simulation (red solid line with shaded area).The rate profile is the same as the magenta curve in Fig. 2c, where η c = 0.05 a.u. and τ c = 1000 fs.The IR spectra are simulated by HEOM, with details presented in Supplementary Note 6.The optical spectra can be well approximated as A ν ðω À ω 0 Þ in Eq. ( 14) (red open circles), which is visually identical to the HEOM results.The similar trend of the vibration spectra for the molecular system outside the cavity and the VSC-modified rate constant profile are a ubiquitous feature for most of the VSC experiments so far 1,5,10,14 , with the peaks both located at ω c = ω 0 and the widths roughly at Γ ν (although there are counter-examples, such as Fig. 3a of ref. 2).This feature is observed in current numerical simulations, as well as in the previous work 35 , which can be explained by the kVSC expression in Eq. (17).

FGR rate theory in the intermediate regime Under the intermediate regime (τ c ∼ Ω À1
R ), it is difficult to have a simple reaction mechanism and derive an analytical rate constant expression.This is indeed the case for Kramers turnover when the friction parameter is in between the energy and spatial diffusion limits 38 .A similar situation also occurs for the theory of electron transfer under the non-adiabatic limit (golden rule, Marcus Theory) or adiabatic limit (Born-Oppenheimer, Hush Theory), where well-defined rate theories are available in both regimes [52][53][54][55] , but there is no analytic theory for the entire crossover region.Nevertheless, one can apply an ad hoc approach by interpolating the two FGR expressions in Eqs. ( 2) and ( 17) as follows 53,55 which is the second main theoretical result in this work.The numerical result of FGR rates using k int VSC in Eq. ( 20) is presented in Fig. 2b (golden dashed line), with a re-scaling factor of 0.5 applied to both k VSC (Eq.( 10)) and kVSC (Eq.( 17)).One can see that Eq. ( 20) correctly captured the turnover behavior in the τ c -dependence of VSC rate constant, which maximizes at around τ c = 200 fs and agrees with the HEOM simulations, although being less accurate than either Eq.(2) in the lossy regime or Eq. ( 17) in the lossless regime.As a corollary of k int VSC , the maximum enhancement of the VSC rate constant can be reached when τ c ¼ Ω À1 R .This is because under the resonance condition ω c = ω 0 , Eq. ( 20) becomes (c.f.Eqs. ( 9) and ( 19)) where the equal sign is satisfied under τ c ¼ Ω À1 R .Figure 3b presents the τ c -dependence of VSC rate constants under different Rabi splittings Ω R , obtained from the numerically exact HEOM  17) (blue thick line), as well as the IR spectra of the bare molecule system from HEOM (thick solid line) and using Eq. ( 14) (open circles).The rate profile is the same as the violet curve in Fig. 2c.b Cavity lifetime τ c -dependence of the VSC rate constant k/k 0 under various Ω R obtained from HEOM simulations (open circles with thin guiding line) as well as the interpolated expression in Eq. ( 20) (solid lines), and the cavity frequency is fixed at the resonance condition ω c = ω 0 .The dashed vertical lines denote the position where τ c ¼ Ω À1 R .c Relation between k/k 0 at resonance (ω c = ω 0 ) and the Rabi splitting Ω R , with results obtained from HEOM (red circles) and FGR (red solid line) using kVSC in Eq. (19).The change of the effective free energy barrier height Δ(ΔG ‡ ) is also presented, with HEOM (blue circles) and the FGR (blue solid line) using Eq.(22).simulations (open circles with thin guiding line) as well as the interpolated expression in Eq. ( 20) (solid lines).All pairs of curves show a similar turnover behavior along τ c but differ in the peak positions.The dashed vertical lines denote the position where τ c ¼ Ω À1 R at the corresponding Ω R value, which coincides with the peak positions of the turnover curves.As a result, the expression of k int VSC predicts that the maximum enhancement of VSC rate constants is reached when τ c ¼ Ω À1 R , in agreement with the numerically exact simulations.

Effect of the Rabi splitting
We further explore the effect of the light-matter coupling strength on the VSC rate constant and the accuracy of the FGR expression in Eq. ( 19) (Eq.( 17) under the resonance condition) in the lossless regime.By doing so, we fix the cavity lifetime as τ c = 1000 fs. Figure 3c presents the relation between k/k 0 at resonance (ω c = ω 0 ) under various Rabi splitting Ω R , obtained from the HEOM simulations (red circles) and the FGR expression (red solid line) using kVSC in Eq. ( 19) with a re-scaling factor of 0.5 on kVSC .Over up to 100 cm −1 Rabi splitting, the FGR expression (Eq.( 19)) correctly captures the Ω R -dependence that first scales as kVSC / Ω 2 R , then plateau (saturated).This is because when Ω R becomes large, only k1 (Eq.( 12)) is rate limiting, which is Ω R -independent.
Figure 3c further presents the change of the effective free energy barrier Δ(ΔG ‡ ), directly calculated from the rate constant ratio k/k 0 obtained from HEOM simulations.To account for the "effective change" of the Gibbs free energy barrier Δ(ΔG ‡ ) as follows 4,11,36 ΔðΔG Note that this is not an actual change in the free-energy barrier, but rather an effective measure of the purely kinetic effect.Here, one can see a non-linear relation of Δ(ΔG ‡ ) with Ω R that has been observed experimentally 6 , and the theory in Eq. ( 19) provides a semi-quantitative agreement with the numerically exact results from HEOM simulations.
Temperature dependence of the VSC rate constant Experimentally, it was found that VSC induces changes in both effective activation enthalpy and activation entropy when using the Eyring equation to interpret the change of the rate constant 1,10,11 , which remains to be theoretically explained.We emphasize that based on our current theory, the VSC modification mechanism is not due to the direct modification of the Entropy or Enthalpy, but rather through the mechanisms summarized in Eq. ( 11) (for the lossless regime) and Eq. ( 1) (for the lossy regime).However, if one chooses to interpret the change of the rate constant through these enthalpy and entropy changes, then the current theory in Eq. ( 17) can indeed explain both changes.Using the Eyring equation, the temperature dependence of the reaction rate constant is where ΔH ‡ and ΔS ‡ are the effective activation enthalpy and entropy, respectively, which can be extracted by plotting lnðk=TÞ as a function of 1/T.We further denote the effective activation enthalpy and entropy inside the cavity as ΔH z c and ΔS z c , respectively, and the corresponding values outside the cavity as ΔH z 0 , ΔS z 0 , respectively.One can further define their difference as ΔΔH z ΔH z c À ΔH z 0 , and ΔΔS z ΔS z c À ΔS z 0 , which characterizes the pure cavity induced effects.According to the assumption that k = k 0 + k VSC , they can be evaluated analytically as follows where the detailed derivations are provided in Supplementary Note 7. Eq. 24a-b can be evaluated by using HEOM results, or the FGR expressions, either k VSC in Eq. ( 2) (or Eq. ( 10) to be more accurate) for the lossy case or kVSC in Eq. ( 17) for the lossless case.The previous work based on the classical Grote-Hynes rate theory 15 can only explain the change in ΔΔS ‡ .The current FGR-based theory can explain changes in both ΔΔH ‡ and ΔΔS ‡ , which has been observed in experiments 4,6 .Figure 4a presents the temperature dependence of the VSC rate constant, plotting as lnðk=k 0 Þ as a function of 1/T.The cavity lifetime is fixed as τ c = 1000 fs, and the cavity frequency is ω c = ω 0 .Figure 4a shows the Eyring-type plots for reactions outside the cavity (black points) and inside a resonant cavity under various light-matter coupling strengths.The rate constants were obtained from HEOM simulations (dots), and fitted by the least square to obtain linearity (thin lines).One can see that as Ω R increases, the slope of the Eyring plots becomes more negative (an increasing the effective activation enthalpy).Meanwhile, the effective entropy also increased significantly as one increase Ω R .The current theory explains both changes of ΔΔH ‡ and ΔΔS ‡ , and the temperaturedependence in Fig. 4a has been experimentally observed (e.g., Fig. 4 in ref. 56).
Figure 4b presents the change of the effective activation enthalpy ΔΔH ‡ as increasing Ω R .The HEOM results for ΔΔH ‡ (blue open circles with thin guiding lines) are extracted from the slopes of the fitted lines in Fig. 4a.Further, the FGR results (gold dashed line) are presented, in which k VSC is calculated using Eq. ( 17) (re-scale by a factor of α = 0.5) and plug-in Eq. (24a) to obtain the cavity induced change ΔΔH ‡ .When k VSC is small,  17) (where the value of kVSC in Eq. ( 17) is re-scaled by a factor of 0.5).c Effective activation entropy under different Ω R values obtained from the exact HEOM simulations (blue open circles) and the FGR results (gold dashed line) using Eq. ( 17) (where the value of kVSC in Eq. ( 17) is re-scaled by a factor of 0.5).
Eq. ( 24a) is proportional to k VSC , i.e., proportional to Ω 2 R according to the analytic FGR rate theory (see Eq. ( 17)).One can see that from Fig. 4b when Ω R < 15 cm −1 , ΔH z c increases quadratically with Ω R , and the FGR results agree with the HEOM results.When Ω R > 15 cm −1 , the behavior deviates from quadratic scaling, and FGR results still closely match the trend of HEOM results.Figure 4c presents the change of the effective activation entropy ΔΔS ‡ , with results obtained from HEOM (blue open circles with thin line) and FGR (golden dashed line), where the FGR also provides a good agreement with the exact results.Note that Eq. 24a-b also works well in the lossy regime, where the results with τ c = 100 fs and k/k 0 evaluated using Eq. ( 10) are presented in Supplementary Fig. 3.

Resonance condition at the normal incidence
The dispersion relation of a Fabry-Pérot (FP) microcavity 6,16,57 is where c is the speed of light in vacuum, n c is the refractive index inside the cavity, c/n c is the speed of the light inside the cavity, and θ is the incident angle, which is the angle of the photonic mode wavevector k relative to the norm direction of the mirrors.For simplicity, we explicitly drop n c throughout this paper (because of the experimental value n c ≈ 1).The many-mode Hamiltonian is provided in Supplementary Note 8.When k ∥ = 0 (or θ = 0), the photon frequency is which is the cavity frequency we introduced in the previous discussions (Eq.( 4)).Experimentally, it is observed that only when ω c = ω 0 (known as the normal incidence condition) will there be VSC effects 2,4,6,10,31 .For a reddetuned cavity (ω c < ω 0 ), there are still a finite number of modes (with a finite value of k ∥ ), such that ω k = ω 0 .This is referred to as the oblique incidence, but there is no observed VSC effect even though polariton states are formed 1,6,31 .
As mentioned in ref. 2, "Tuning the cavity so that VSC occurs at normal incidence is essential to observe the modification of chemical properties.In this condition, the system is at the minimum energy in the polaritonic state 2 ".Despite recent theoretical progress 58,59 , there is no accepted theoretical explanation for VSC effect only observed at the normal incidence, although intuitively, the group velocity ∂ω k /∂k ∥ = 0 at k ∥ = 0 and make that point special, as hinted by Ebbesen and co-workers 2 .Our recent work suggests that for the analytic expression k VSC (Eq.( 2)), it is possible to explain such a normal incidence effect when considering many cavity modes 45 .In this work, we theoretically explore such normal incidence conditions for the new analytic expression kVSC (Eq.( 17)) under the lossless regime.
For k ∥ > 0, the mode has a finite momentum in the in-plane direction.Because of this in-plane propagation, the photon leaving the effective mode area is characterized by the following effective lifetime 45 where D characterizes the spatial extent of a given mode (along the k ∥ direction).Using the experimental molecular density and the effective number of molecules coupled to a given mode, one can estimate D ≈ 10 À1 ∼ 100 μm, with details provided in Supplementary Note 10, section A. We want to emphasize that τ ∥ differs from the cavity lifetime τ c which considers the photon loss in the k ⊥ direction due to leaking outside the cavity.As a result of τ ∥ , the thermal photon number should be modified as 45 n(ω k ) → n eff (ω k ) with the following expression due to the detailed balance relation.
Using the same procedure of the FGR derivation (as used for Eq. ( 17)), the VSC enhanced rate constant under the lossless regime is where g c ¼ μ LL 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=ð2_ϵ 0 VÞ p is the Jaynes-Cummings 60 type light-matter coupling strength that does not depend on ω k .Note that in the literature, Ω R ¼ 4g 2 c ω c for the resonance condition.When there is only one mode, Eq. ( 29) reduces back to Eq. (17).Further, ϕ k describes the angle between the molecular dipole and the k th cavity mode.For the 1D FP cavity (one dimensional for the k ∥ direction), cos ϕ k ¼ 1.For the 2D FP cavity (two dimensional for the k ∥ direction), we assume an isotropic average cos 2 ϕ k !hcos 2 ϕ k i ¼ 1=2.As such, the rate constant in Eq. ( 29) can be evaluated by replacing the summation with an integral as follows where g D (ω) is the DOS for the cavity modes.Using the cavity dispersion relation in Eq. ( 25), the photonic density of states (DOS) for a 1D FP cavity is expressed as follows 45 , where Θ(ω − ω c ) is the Heaviside step function, Δk ∥ is the spacing of the in-plane wavevector k ∥ (or the k-space lattice constant).Note that g 1D (ω) has a singularity at ω = ω c , which is known as (the first type of) the van-Hove-type singularity 61 .The DOS for a 2D FP cavity is expressed as 45 which does not have any singularity.For a 1D FP cavity, using g 1D (ω) in Eq. ( 31) and evaluating the integral in Eq. (30) (see details in Supplementary Note 9) results in which is identical to Eq. ( 17) (with Ω R ¼ 4g 2 c ω c ), with an additional M ¼ R g 1D ðωÞdω which is the number of cavity modes.Thus, for a 1D FP cavity, the peak of the expression in Eq. ( 33) is located at ω c = ω 0 where k ∥ = 0, due to the presence of the van-Hove singularity.This means that VSC modification occurs only when ω c = ω 0 for a 1D FP cavity.We have also numerically evaluated Eq. ( 30) and compared it with Eq. ( 33) for the VSCmodified rate constant, presented in Supplementary Fig. 4, which shows a nearly identical behavior.
On the other hand, all of the known VSC experiments 1,6,10,31 have been performed in 2D FP cavities.For a 2D FP cavity, using g 2D (ω) in Eq. ( 32) to evaluate Eq. ( 29), the VSC rate constant becomes is the number of modes, ω m is the integration cutoff frequency (which is treated as a convergence parameter), and τ k ðωÞ ¼ ðD=cÞ Á ω= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω 2 À ω 2 c p (c.f.Eq. ( 27)).See Supplementary Note 9 for detailed derivations.As a crude estimation, one can approximate A ν ðω À ω 0 Þ ≈ δðω À ω 0 Þ, such that the integral in Eq. ( 34) can be evaluated analytically, leading to Since usually cτ c =D ≫ 1, Eq. ( 35) has a sharp maximum value at ω c = ω 0 and tails toward the ω c < ω 0 side.Figure 5 presents the VSC-enhanced rate constant using the FGR expression under different Rabi splitting Ω R values inside (a) a 1D FP cavity and (b) a 2D FP cavity, where the cavity lifetime is τ c = 1000 fs. Figure 5a presents k=k 0 ¼ 1 þ 0:5 k1D VSC =Mk 0 , where the number of modes M has been divided to present a normalized result.This is identical to the singlemode expression (Eq.( 17)) due to the van-Hove-type singularity in the 1D DOS (see Eq. ( 31)). Figure 5b presents k=k 0 ¼ 1 þ 0:5 k2D VSC =k 0 value for a single molecule coupled to many modes inside a 2D FP cavity, where k2D VSC was evaluated by performing direct sum using Eq. ( 29) (solid lines), as well as by using k 2D VSC expression in Eq. ( 34) (open circles), which are identical to each other.Note that both k1D VSC and k2D VSC are re-scaled by a factor of 0.5 to be consistent with Fig. 2. Here, we choose D ¼ 1 μm for the effective mode diameter, the effective cavity size L ¼ 1 mm (probing area) to discretize the 2D cavity dispersion relation when using Eq. ( 29) (solid lines), with ω m = 5ω c which generates a total number of M≈10 6 modes for the 2D FP cavity.We use the same ω m value to perform integration using Eq. ( 34) (open circles).The details are provided in Supplementary Note 10, section B. One can observe that the resonance peak is still centered around ω c = ω 0 but slightly red-shifted, demonstrating the normal incidence condition.The approximate analytic expression of k2D VSC in Eq. ( 35) gives a similar long tail for ω c < ω 0 but a much sharper decay for ω c ≥ ω 0 .Overall, the resonance peak is asymmetric as it tails toward the lower energy regions.Future VSC experiments are required to explore if there is any asymmetry in the rate constant profile.

Experimental connections
The current theory is valid for N = 1 or a few molecules strongly coupled to the cavity, such that the individual light-matter coupling η c is strong.Experimentally, it is now possible to achieve strong (or even ultra-strong 62 ) light-matter couplings between a plasmonic nanocavity and a few vibrational modes 62,63 , such that Ω R ≫ τ À1 c (for N = 1).In these experimental setups 62,63 , the current theory ( kVSC in Eq. ( 17)) can be directly applied, and all of the predictions could be verified experimentally, e.g., the τ c behavior in Fig. 2 and various scaling relations in Fig. 3.
On the other hand, in all existing VSC experiments 1,2,10,14 , the Rabi splitting is achieved through a collective light-matter coupling between N vibrational modes with the cavity, such that Eq. ( 8) should be modified as 16,31,64,65 It was estimated that N ≈ 10 6 ~10 12 per effective cavity mode 64 , and Ω R,N ≈ 100 cm −1 for the typical VSC experiments 4,10 .The strong coupling condition in the experiments is achieved when Ω R;N ≫ τ À1 c and the optical spectra of the molecule-cavity hybrid system have a peak splitting.However, the fundamental mechanism of the experimentally observed VSC effect (which happens under the collective coupling regime, Eq. ( 36)) remains to be explained.
If all molecules are perfectly aligned with the cavity field, the coupling strength per molecule η c is bound to be very weak ( ∼ Ω R;N = ffiffiffiffi N p ).Recent theoretical work 66 suggests that disorders of the molecular dipole distribution along the field polarization will create local strong coupling spots 66 , and in these "hot spots", only a few molecules are strongly coupled to the cavity 66 (which resembles a form of spin glass).If this is the case in the VSC experiments, then combining the kVSC in Eq. ( 17) with the disorderenhanced local coupling theory in Ref. 66 would likely explain the VSC enabled effect.On the other hand, the VSC-induced rate constant changes could originate from a non-trivial collective effect even though the individual η c is tiny 31 .In this case, one has the scenario that Ω R ≪ τ À1 c (where Ω R ~ηc ) but Ω R;N ≫ τ À1 c (due to the large N).As such, one would expect to use the k VSC (Eq.(2) or Eq. ( 10)) to describe the rate constant associated with a single molecule, add up all contributions in FGR and normalize it with 1/N (to avoid a simple concentration effect).This, however, will not give any significant change in the VSC-modified rate constant 45 , due to the large 1/N normalization factor.Future work needs to address this challenge, which might emerge from non-trivial collective effect due to non-local collective light-matter coupling 67,68 .
Nevertheless, our current theory suggests that measuring the τ cdependence of k/k 0 could be the key to unraveling the fundamental mechanism in VSC.For example, under the strong coupling regime, if k/k 0 decreases as τ c increases, then the mechanism is likely to be kVSC in Eq. ( 17) with the disorder-enhanced local coupling theory 66 .On the other hand, under the strong coupling condition Ω R;N ≫ τ À1 c , if k/k 0 increases as τ c increases, then it implies that under the single molecule level Ω R ≪ τ c , and the VSC mechanism is likely to be k VSC (Eq.(2) or Eq. ( 10)) with a collective mechanism yet to be discovered.Experimentally, the cavity lifetime (or the quality factor) for the distributed Bragg reflector (DBR) FP cavity can be modified by changing the number of coating layers 69 or the curvature of the mirrors 70 .Other possible cavity structures that could achieve various Q factors are the "open" photonic structures 71,72 , which might be more suitable for polaritonic chemistry than planar cavities, as these mirror-free open structures generally support lower quality photonic modes than the standard FP cavity design.In either case, experimental measurements on the VSC (Eq.( 33)) where the number of modes M is divided, which is identical to the single mode case in Eq. ( 17).b FGR rate profiles for many mode cases inside a 2D FP cavity, where the results obtained by performing direct sum using Eq. ( 29) (solid lines) and by performing integration using k2D VSC in Eq. ( 34) (open circles) are presented.
cavity lifetime dependence of the rate constant will provide invaluable insights into the nature of the VSC effects.
Further, going back to the experimental details, when comparing the theoretical results of k/k 0 (cavity effect for inside and outside the cavity) to experiments, one will need to compare the results for two systems with different cavity lifetimes, and the outside cavity case could be the experimental set up of FP cavity with a τ c → 0 limit (as our theoretical results suggested).This is because, in the real world, all planar wavelength scale structures support well-defined photonic modes.A final note is that many existing experiments compare reaction rate constants for molecules inside a cavity and outside a cavity (or on-resonance cavity and off-resonance cavity).If the on-resonance cavity sample has a different reaction rate to the non-cavity and/or off-resonance cavity sample, it is assumed that this change must be caused by strong coupling.This is a strong assumption and, if incorrect, could easily lead to false positives 21,73 .More careful experiment designs are needed in the future to differentiate between changes in reaction rate constant caused by polaritonic and non-polaritonic effects 21,73 .As pointed out by Thomas and Barnes 74 , it is also possible that cognitive bias could significantly influence the interpretation of strong coupling experiments and caution is needed to prevent false positive results.

Conclusion
We developed an analytic theory for the VSC-modified rate constant kVSC (Eq.( 17)) for a single molecule strongly coupled to the cavity, under the lossless regime (when τ À1 c ≪ Ω R ).This analytic theory is based on the mechanistic observation of sequential rate-determining steps |ν L ; 0 !|ν L ; 1 !|ν 0 L ; 0 (outlined in Eq. ( 11)), which are observed in our numerically exact quantum dynamics simulations (see Supplementary Note 4).The theory kVSC (Eq.( 17)) explains the resonance condition ω c = ω 0 and the close connection between the rate constant modification kVSC and the optical lineshape A ν ðω À ω 0 Þ (Eq.( 14)).This explains why the VSC-modified rate distribution closely follows the optical spectra as observed in the VSC experiments 1,2,10,14 .This analytic theory kVSC provides accurate ω c -dependence of the VSC rate constant enhancement compared to the numerically exact results from HEOM simulations.
The current analytic theory kVSC (Eq.( 17)) also explains why under the lossless regime (Ω R ≫ τ À1 c ), the rate constant increase when decreasing the cavity lifetime τ c (see Eq. ( 19)), agreeing with the previous numerically exact simulations 35 .Under the lossy regime (Ω R ≪ τ À1 c ), our previous work 36 provides an analytic theory k VSC (Eq.( 2)), which predicts that the rate constant will increase as τ c increases (see Eq. ( 9)).Both k VSC and kVSC agree well with the numerical exact HEOM results under their specific regimes.The combination of kVSC (Eq.( 17)) and k VSC (Eq.( 2)) provides a complete picture of the τ c -dependence of the VSC rate constant modification and suggests there should be a turnover behavior.The physical picture of the cavity enhancement effect for the rate constant is clarified by the reaction mechanisms in Eq. (1) (limited by vibrational excitation) under the lossy regime and Eq. ( 11) (limited by photonic excitation) under the lossless regime.The cavity loss parameter τ À1 c can thus be viewed as a friction parameter associated with the cavity mode qc and the turnover behavior of the rate constant is essentially the Kramers turnover.We also provided an interpolating scheme (Eq.( 20)) for the description of the turnover phenomenon and predicted that the maximal enhancement will be reached when τ c ¼ Ω À1 R (see Eq. ( 21)), all agree well with the exact simulations.The analytic theory kVSC (Eq.( 17)) predicts that the VSC rate enhancement (Eq.( 19)) scales as kVSC =k 0 / Ω 2 R as the light-matter coupling increases, then plateaus when Ω R becomes large.This is in excellent agreement with the numerically exact HEOM simulations and provides a non-linear relation between the change of the effective free energy barrier and the light-matter coupling strength (Fig. 3c), which has been observed in the VSC experiments 4,6 .The theory kVSC (Eq.( 17)) also predicts changes in both effective activation enthalpy and entropy (as observed in the experiments 6 ), which agrees well with the numerical exact HEOM results within all the parameter regimes we explored.Although a re-scale parameter α = 0.5 is needed to bring the numerical values of the FGR rate constants in consistency with the HEOM exact results, the overall scaling relations with respect to Ω R , ω c , τ c , and T reported in the paper are rather general and impressive, which should not be restricted to the detailed shape of the potential energy surface or environmental spectral density functions.We further generalized the kVSC expression to consider the many mode effects, and the resulting theories (Eq.( 33) for 1D FP cavity and Eq. ( 34) for 2D cavity) predict the normal-incidence resonance condition: the peak of the rate constant enhancement occurs when k ∥ = 0 and ω c = ω 0 .Last but not least, our current theory also predicts that for two chemically similar reactions, if one satisfies k 1 ≪ k 2 , k 3 but the other does not (due to the low reaction barrier), then there will be a VSC effect for the first reaction but not for the second one.This is because for the second reaction, |ν L ! |ν 0 L is no longer rate limiting, and the cavity modification of this process will no longer influence the apparent rate constant.This might be the explanation for the recently observed null effects in VSC experiments 22,23 .
Despite the successes of the theory, it is limited to the situation of a single molecule strongly coupled to the cavity.In most of the experiments, a large collection of molecules (N = 10 6 ~10 12 ) are collectively coupled to the cavity, and the coupling strength per molecule is rather weak.In the future, we aim to generalize the current analytic rate constant expression to explain the resonance suppression and the collective effect, to build a unified theory for the VSC-modified rate constant.Future efforts shall be focused on applying the current simulation approach and theory to realistic reaction systems with atomistic details 26,75 .

Model Hamiltonian
We use a double-well (DW) potential to model the ground state chemical reaction 76,77 Vð where M is chosen as the proton mass, ω b = 1000 cm −1 is the barrier frequency, and E b = 2120 cm −1 is the barrier height.For the matter Hamiltonian ĤM ¼ T þ V, the vibrational eigenstates |ν i and eigenenergies E i are obtained by solving ĤM |ν i ¼ E i |ν i numerically using the discrete variable representation (sinc-DVR) basis 78 with 1001 grid points in the range of [ − 2.0, 2.0].To facilitate the mechanism analysis, we diabatize the two lowest eigenstates and obtain two energetically degenerate diabatic states both with energies of E ¼ ðE 1 þ E 0 Þ=2 and a small tunneling splitting of Δ = (E 1 − E 0 )/2 ≈ 1.61 cm −1 .Similarly, for the vibrational excited states f|ν 2 ; |ν 3 g, we diabatize them and obtain the first excited diabatic vibrational states in the left and right wells as follows with degenerate diabatic energy of E 0 ¼ ðE 3 þ E 2 Þ=2 and a tunneling splitting of Δ0 ¼ ðE 3 À E 2 Þ=2≈64:05 cm À1 .A schematic representation of these diabatic states are provided in Fig. 1a.Based on the two diabatic states |ν L and |ν 0 L in the left well, we define the quantum vibration frequency of the reactant as which is directly related to the quantum transition of |ν L ! |ν 0 L .Further, Ĥν in Eq. ( 5) is the system-bath Hamiltonian that describes the linear coupling between reaction coordinate R and its phonon bath,

Fig. 2 |
Fig. 2 | Effect of cavity lifetime τ c on the VSC-modified rate constant.Comparisons are made between the numerically exact HEOM results (open circles with thin guiding lines) and the FGR rate constants (both k VSC and kVSC ) which are re-scaled by a factor of 0.5 (solid lines).The light-matter coupling strength is fixed at η c = 0.05 a.u. a Resonance peaks of k/k 0 for VSC effect under the lossy regime (τ c ≪ Ω À1 R ).The FGR rates using k VSC in Eq. (10) (thick solid lines) are compared to the HEOM results (open circles with thin guiding lines) under a variety of τ c values.b The values of k/k 0 under the resonance condition ω c = ω 0 as a function of τ c .The results of HEOM (blue open circles), FGR rates using k VSC in Eq. (10) (red solid line), FGR rates using kVSC in Eq. (17) (blue solid line), and FGR rates using k int VSC in Eq. (20) (gold dashed line) are presented.c Resonance peaks of k/k 0 for VSC effect under the lossless regime (τ c ≫ Ω À1 R ).The FGR rates using kVSC in Eq. (17) (thick solid lines) are compared to the HEOM results (open circles with thin guiding lines) under a variety of τ c values.

Fig. 3 |
Fig. 3 | Influence of cavity frequency, cavity lifetime, and light-matter coupling strength on kVSC .a The rate profile k/k 0 obtained from HEOM simulations (blue open circles) and the FGR expression using Eq.(17) (blue thick line), as well as the IR spectra of the bare molecule system from HEOM (thick solid line) and using Eq.(14) (open circles).The rate profile is the same as the violet curve in Fig.2c.b Cavity lifetime τ c -dependence of the VSC rate constant k/k 0 under various Ω R obtained from HEOM simulations (open circles with thin guiding line) as well as the

Fig. 4 |
Fig. 4 | Temperature dependence of the VSC rate constant.The cavity lifetime τ c is fixed at 1000 fs, and the cavity frequency is kept at the resonance condition ω c = ω 0 .a Eyring-type plots for lnðk=k 0 Þ as a function of 1/T, for reaction outside the cavity (black points) and inside the resonant cavity under various light-matter coupling strengths.b Effective activation enthalpy under different Ω R values, with the results obtained from the exact HEOM simulations (blue open circles) and the FGR results (gold dashed line) using Eq.(17) (where the value of kVSC in Eq. (17) is re-scaled by a factor of 0.5).c Effective activation entropy under different Ω R values obtained from the exact HEOM simulations (blue open circles) and the FGR results (gold dashed line) using Eq.(17) (where the value of kVSC in Eq. (17) is re-scaled by a factor of 0.5).

Fig. 5 |
Fig.5| Normal Incidence effect.FGR rate profiles of k/k 0 as a function of ω c , where the cavity lifetime is τ c = 1000 fs.Results under various light-matter coupling strengths are presented.a FGR rate using k1D VSC (Eq.(33)) where the number of modes M is divided, which is identical to the single mode case in Eq. (17).b FGR rate profiles for many mode cases inside a 2D FP cavity, where the results obtained by performing direct sum using Eq.(29) (solid lines) and by performing integration using k2D VSC in Eq. (34) (open circles) are presented.