Recent experiments demonstrate polaritons under the vibrational strong coupling (VSC) regime can modify chemical reactivity. Here, we present a complete theory of VSCmodified 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 VSCmodified 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 inplane momentum inside a FabryPérot cavity.
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Abstract 
Abstract We present a theory that explains the resonance effect of the vibrational strong coupling (VSC) modified reaction rate constant at the normal incidence of a Fabry–Pérot (FP) cavity. This analytic theory is based on a mechanistic hypothesis that cavity modes promote the transition from the ground state to the vibrational excited state of the reactant, which is the ratelimiting step of the reaction. This mechanism for a single molecule coupled to a singlemode cavity has been confirmed by numerically exact simulations in our recent work in [J. Chem. Phys. 159, 084104 (2023)]. Using Fermi’s golden rule (FGR), we formulate this rate constant for many molecules coupled to many cavity modes inside a FP microcavity. The theory provides a possible explanation for the resonance condition of the observed VSC effect and a plausible explanation of why only at the normal incident angle there is the resonance effect, whereas, for an oblique incidence, there is no apparent VSC effect for the rate constant even though both cases generate Rabi splitting and forming polariton states. On the other hand, the current theory cannot explain the collective effect when a large number of molecules are collectively coupled to the cavity, and future work is required to build a complete microscopic theory to explain all observed phenomena in VSC.
Free, publiclyaccessible full text available February 23, 2025 
We present numerically exact quantum dynamics simulations using the hierarchical equation of motion approach to investigate the resonance enhancement of chemical reactions due to the vibrational strong coupling (VSC) in polariton chemistry. The results reveal that the cavity mode acts like a “ratepromoting vibrational mode” that enhances the ground state chemical reaction rate constant when the cavity mode frequency matches the vibrational transition frequency. The exact simulation predicts that the VSCmodified rate constant will change quadratically as the light–matter coupling strength increases. When changing the cavity lifetime from the lossy limit to the lossless limit, the numerically exact results predict that there will be a turnover of the rate constant. Based on the numerical observations, we present an analytic rate theory to explain the observed sharp resonance peak of the rate profile when tuning the cavity frequency to match the quantum transition frequency of the vibrational ground state to excited states. This rate theory further explains the origin of the broadening of the rate profile. The analytic rate theory agrees with the numerical results under the golden rule limit and the short cavity lifetime limit. To the best of our knowledge, this is the first analytic theory that is able to explain the sharp resonance behavior of the VSCmodified rate profile when coupling an adiabatic ground state chemical reaction to the cavity. We envision that both the numerical analysis and the analytic theory will offer invaluable theoretical insights into the fundamental mechanism of the VSCinduced rate constant modifications in polariton chemistry.

We generalize the quasidiabatic (QD) propagation scheme to simulate the nonadiabatic polariton dynamics in molecule–cavity hybrid systems. The adiabaticFock states, which are the tensor product states of the adiabatic electronic states of the molecule and photon Fock states, are used as the locally welldefined diabatic states for the dynamics propagation. These locally welldefined diabatic states allow using any diabatic quantum dynamics methods for dynamics propagation, and the definition of these states will be updated at every nuclear time step. We use several recently developed nonadiabatic mapping approaches as the diabatic dynamics methods to simulate polariton quantum dynamics in a Shin–Metiu model coupled to an optical cavity. The results obtained from the mapping approaches provide very accurate population dynamics compared to the numerically exact method and outperform the widely used mixed quantumclassical approaches, such as the Ehrenfest dynamics and the fewest switches surface hopping approach. We envision that the generalized QD scheme developed in this work will provide a powerful tool to perform the nonadiabatic polariton simulations by allowing a direct interface between the diabatic dynamics methods and ab initio polariton information.

We derive a rigorous nuclear gradient for a moleculecavity hybrid system using the quantum electrodynamics Hamiltonian. We treat the electronic–photonic degrees of freedom (DOFs) as the quantum subsystem and the nuclei as the classical subsystem. Using the adiabatic basis for the electronic DOF and the Fock basis for the photonic DOF and requiring the total energy conservation of this mixed quantum–classical (MQC) system, we derived the rigorous nuclear gradient for the molecule–cavity hybrid system, which is naturally connected to the approximate gradient under the Jaynes–Cummings approximation. The nuclear gradient expression can be readily used in any MQC simulations and will allow one to perform the nonadiabatic onthefly simulation of polariton quantum dynamics. The theoretical developments in this work could significantly benefit the polariton quantum dynamics community with a rigorous nuclear gradient of the molecule–cavity hybrid system and have a broad impact on the future nonadiabatic simulations of polariton quantum dynamics.more » « less

We present the rigorous theoretical framework of the generalized spin mapping representation for nonadiabatic dynamics. Our work is based upon a new mapping formalism recently introduced by Runeson and Richardson [J. Chem. Phys. 152, 084110 (2020)], which uses the generators of the [Formula: see text] Lie algebra to represent N discrete electronic states, thus preserving the size of the original Hilbert space. Following this interesting idea, the Stratonovich–Weyl transform is used to map an operator in the Hilbert space to a continuous function on the SU( N) Lie group, i.e., a smooth manifold which is a phase space of continuous variables. We further use the Wigner representation to describe the nuclear degrees of freedom and derive an exact expression of the timecorrelation function as well as the exact quantum Liouvillian for the nonadiabatic system. Making the linearization approximation, this exact Liouvillian is reduced to the Liouvillian of several recently proposed methods, and the performance of this linearized method is tested using nonadiabatic models. We envision that the theoretical work presented here provides a rigorous and unified framework to formally derive nonadiabatic quantum dynamics approaches with continuous variables and connects the previous methods in a clear and concise manner.more » « less

We derive the $\mathcal{L}$MFE method to incorporate Lindblad jump operator dynamics into the meanfield Ehrenfest (MFE) approach. We map the density matrix evolution of Lindblad dynamics onto pure state coefficients using trajectory averages. We use simple assumptions to construct the $\mathcal{L}$MFE method that satisfies this exact mapping. This establishes a method that uses independent trajectories which exactly reproduces Lindblad decay dynamics using a wavefunction description, with deterministic changes of the magnitudes of the quantum expansion coefficients, while only adding on a stochastic phase. We further demonstrate that when including nuclei in the Ehrenfest dynamics, the $\mathcal{L}$MFE method gives semiquantitatively accurate results, with the accuracy limited by the accuracy of the approximations present in the semiclassical MFE approach. This work provides a general framework to incorporate Lindblad dynamics into semiclassical or mixed quantumclassical simulations.more » « less