Fig. presents the polariton dynamics of the SM2 model coupled to a cavity. This model has been used to investigate how cavity can influence proton-coupled electron transfer reaction with the exact factorization approach. The PES of SM2 is asymmetric and there is an avoid crossing near R = 2.0 a.u.. The photon energy we use is 0.1 a.u., which causes a light- When the coupling is weak (g c = 0.001), the polaritonic PESs (Fig. ) have nearly identical curvatures of the photon dressed states |g1 and |e0 (which is indicated by the color coding of the surfaces), except at R ≈ -4 a.u. where the |e, 0 and |g, 1 states mix. In Fig. and Fig. , even for a small light-matter coupling strength g c = 0.001, the system will have a finite population for |g0 state. This is different compared to the case of SM1 (Fig. and Fig. ), where the dynamics is largely confined with in the JC subspace {|g1 , |e0 } under the same coupling strength. Although SM2 has a large permanent dipole, the population of |g0 starts to grow later in time (20 fs) after the initial excitation. The population transfer to these states are mainly due to the electronic NAC d eg that directly couples the |g1 state to |e1 state, as well as the |e0 state to |g0 state. Note that the |e1 state does not have a significant population (see solid lines for the exact results), however, both Ehrenfest dynamics (open circles in Fig. ) and FSSH approach (dots in Fig. ) incorrectly predict a larger |e1 population, due to their less accurate MQC approximation.