In Section 3.1.1, we have described one possible basis for the representation of the PF Hamiltonian (Eq. 13). However, this is not the only choice. In fact, the adiabatic-Fock representation is useful in the weak coupling limit, since the basis is simply the basis that diagonalizes the unperturbed electronic ( Ĥel ) and photonic ( Ĥph ) Hamiltonians. In this sense, F I G U R E 2 (a) Diabatic Mulliken-Hush (MH) potentials V I (R) (red) and V C (R) (blue), with diabatic coupling V IC (gold line). (b) Matrix elements of μ in the adiabatic representation (dashed curves) µ g g (pink), µ ee (cyan), and µ eg (gold), as well as in the diabatic representation (solid lines) µ II (red) and µ CC (blue). Here µ II = µ I (red) and µ CC = µ C in the MH representation since no off-diagonal dipole elements are present in the dipole operator. (c) Polaritonic potentials color-coded according to the number of photons with four relevant avoided crossings labeled as R 0 , R 1 , R 2 , and R 3 . The black solid vertical arrow indicates the initial photoexcitation, and the dashed lines illustrate the dynamics of the hybrid system. (d) Time-dependent photon populations. For panels (c)-(d), the light-matter coupling strength set to A 0 = 0.127 a.u. with a cavity frequency of ω c = 1.5 eV.