where M is the mass of the proton (nucleus in the SM model), R 0 is the position with a minimum potential energy of the ground electronic state. Here, χ(R) is the vibrational ground state wavefunction on the ground electronic states, centered at R 0 under the harmonic approximation, with the harmonic oscillation frequency is ω. For SM1, R 0 = -4.156 and ω = 0.00270 a.u.; for SM2, we use 48 R 0 = -4 and ω = 0.000382 a.u. Further, we assume that the cavity field polarization direction is always aligned with the direction of the dipole operator μ, such that • µ γν = µ γν (for {ν, γ} = {e, g}) where µ γν is the magnitude of μ. Explicitly consider the angle between and μ will generate a polariton induced conical intersection (even for a diatomic molecule), which will induce geometric phase effects.