The effects of changing ω c and g c on the hybrid polariton surfaces can be understood as follows. Changing the cavity frequency ω c will change the energy of a quantum state that has n photons associated with it by the amount ℏω c n. For an electronic transition between a molecular ground and excited state that is coupled to the cavity photon mode, the molecular ground state with n+1 photons (the |g, n+1⟩ state) will couple to the molecule excited state with n photons (the |e, n⟩ state). When the PESs of these ground and excited states have different curvatures, different energetic shifts of ℏω c will cause the |g, n + 1⟩ and |e, n⟩ PESs to intersect at different nuclear configurations. Different points of inter-section (in the nuclear configurational space) lead to different composite curvatures for the upper and lower polariton surfaces which will affect the force the nuclei feel at a given configuration, thus influencing the motion of the nuclear DOFs and altering the reaction pathways compared to the bare molecules outside the cavity. Note that in the above intuitive argument, we have interpreted the Fock state |n⟩ as n photons contained inside the cavity. This is only true when there is no matter inside the cavity, and approximately accurate when the light-matter coupling strength is weak. Rigorously, the photon number operator needs to be gauge transformed as discussed in Eq. 62.