First, we consider the ground state energy of the coupled molecule-cavity hybrid system. Fig. presents a cavity mediated proton-transfer reaction by coupling the aminopropenal molecule to the cavity. This system was recently reported in Ref. 50 utilizing the modified electronic structure methods scQED-HF, scQED-DFT, and scQED-CC to examine the ground state barrier and product energies of the proton reaction. This asymmetric reaction is thus a perfect example to assess the accuracy of the corresponding pQED calculations Fig. presents the potential energy surface (PES) of the reaction outside the cavity, computed using the intrinsic reaction coordinate (IRC) method at the density functional theory (DFT) level (see Theoretical and Computational Methods section). The reaction has a large potential energy barrier around 10 kcal/mol from the reactant (R) to the transition state (TS). This is in nearly perfect agreement with the CC results from Ref. 50. However, the energy difference between the reactant and product from the DFT calculation (∼ 8 kcal/mol) is much larger than the CC results (5 kcal/mol), due to the limitations of DFT. Following the previous work, we define the X-and Ydirection of the molecule, indicated in the inset of Fig. . The Z-direction is perpendicular to the defined X-Y plane. We will place the cavity polarization direction ê will be aligned with either X-or Z-direction of the molecule. Coupling this molecule to an optical cavity can have a strong influence on the PES, depending on the polarization of the cavity with respect to the molecular orientation.