where |Ψ a ⟩ is a single valued polariton adiabatic wavefunction, and Θ(X) is the mixing angle (Eq. 175). The derivation of Eq. 176 can be found in Ref. 95. For a molecule inside a cavity, this Berry phase was analyzed in the work shown in Fig. , adapted from Ref. 95. When a molecule can freely rotate inside a cavity, the angle θ between the dipole of the molecule μ and the cavity polarization ê will change and influence the strength of light-matter coupling (see Fig. ). At a particular bond distance R where the |e, 0⟩ and |g, 1⟩ surfaces intersect, a PICI is formed when the groundto-excited transition dipole moment and cavity polarization vectors are orthogonal (θ = π/2). The nuclear path that takes a particular encirclement around this PICI in the {R, θ} configuration space (Fig. (ii), red path) will gain a phase of π on its adiabatic wavefunction |Ψ a ⟩ after one full encirclement of the PICI based on Eq. 176. On the other hand, taking any other closed path that is not encircling the CI point will not add any additional phase to the wavefunction (Fig. (ii), blue or yellow paths). The effect of this Berry phase can be seen in the probability density of nuclear wavepackets that pass through the PICI point. Note that even though we have used the JC model to intuitively explain the Berry phase effect of the PICI, the actual quantum dynamics simulation was performed using a numerically exact simulation to solve the full PF Hamiltonian (see Eq. 148).