Resonance Theory of Vibrational Strong Coupling Enhanced Polariton Chemistry

We present a complete theory of vibrational strong coupling (VSC) modified rate constants in polariton chemistry when coupling a single molecule to an optical cavity. 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 also explains why VSC rate constant modification closely resembles the optical spectra of the vibration outside the cavity. This analytic expression, together with our previous analytic rate expression under the lossy regime, provides a complete mechanistic picture for the VSC-enhanced rate constant in a simple double well reactive model system. Our analytic theory suggests that there will be a turnover of the rate constant as one changes the cavity lifetime, and the rate constant will first scale quadratically with respect to the light-matter coupling strength and then saturate. The analytic rate constants agree well with the numerically exact hierarchical equations of motion (HEOM) simulations for all explored regimes. Further, we discussed the temperature dependence of the VSC-modified rate constants, where the analytic theory also agrees well with the numerical exact simulations. Finally, we discussed the resonance condition at the normal incidence when considering in-plane momentum inside a Fabry-P\'erot cavity.