Historically, the development of electronic structure methods started in the ground state with HF, DFT, CC, etc. methods and then moved to the excited state with TD-HF, TD-DFT, EOM-CC, etc. The recent development of scQED methods took a similar path, but the production of ground and excited state methods largely overlapped due to the already available electronic structure theory for solving complex many-body Hamiltonians. In the following, we will review some recent studies using scQED as well as pQED schemes, but we will begin with our discussion for the excited state. This is more akin to the original context of quantum optics decades ago, where coupling a cavity to a single atomic transition (ground to excited electronic excitation) was prevalent, as illustrated in the simple features of the Jaynes-Cummings Hamiltonian (Fig. ). Fig. presents a few examples that illustrate the modifications of the excited state electronic structure (or potential energy surfaces) when forming molecular exciton-polaritons. Fig. is one of the first examples 100 of scQED calculations using the equation-ofmotion coupled cluster (EOM-CC) approach (scQED-EOM-CC) to examine the polariton potential energy curve E a (R) (see Eq. 108) of H 2 (left panel) and HF (right panel) when coupled inside a cavity, with the field polarization along the bond axis of each molecule and with a coupling strength λ = 0.05 a.u. The upper (UP) and lower (LP) polaritons are labeled to indicate the location of the main Rabi splitting caused by the |g, 1⟩ and |e, 0⟩ hybridization (as explained by simple JC model in Eq. 6), but the presence of the many-electronic-states and additional many photon-dressed adiabatic states make the UP/LP picture (by the JC model) overly simplified. In Fig. , the blue solid lines indicate the excitonic character, the white solid lines indicate photonic character, and the red dotted lines indicate the original electronic states outside the cavity. Using the photondressed electronic states, one can manipulate and tune the excited state potential energy surfaces to mediate additional transitions or eliminate them. For H 2 , the cavity frequency is close to the first singlet electronic transition (at the Franck-Condon region of the nuclear DOF). The modifications to the excited state PES curvatures can be seen by the induced localization of the UP state, which possesses a minimum near the Frank-Condon point while the original molecular PES (red dashed line) has a purely dissociative character. For the HF molecule coupled to the cavity (right panel), the cavity frequency is near resonant to the ground to the second excited electronic transition. Similar features of the polariton potential can be obtained when this polar molecule (which possesses a permanent ground state dipole) is coupled to the cavity. The authors of that work 100 thus concluded that the permanent dipole does not induce additional interesting effects, which is accurate for this particular system. However, as a reminder of what was discussed previously (Eq. 5), not only the ground state permanent dipole contributes to the light-matter interactions, all excited state permanent and transition dipoles will, in principle, contribute interesting effects. We should note that the energetic alignment of electronic states plays an important role in photophysical properties of polaritons. The commonly used language of "upper polariton" and "lower polariton" could potentially be misleading if the system has many electronic states nearby in energy, as was the case for the H 2 and HF examples presented in this panel. (a) nate near to R CO ≈ 1.35 Å. The shading of the curves in this panel indicates the magnitude of the electronic transition dipole moment between the ground and excited state. At a large cavity frequency ω c = 8.71 eV, the avoided crossing can be reduced by the couplings between the photon-dressed ground state |g, 1⟩ and the higher-energy excited state with zero photons. At a slightly smaller cavity frequency ω c = 8.16 eV, the original potential energy minimum near R CO = 1.5 Å (for the bare molecule) can now be completely removed through the light-matter potential curvature hybridization, tilting the polariton potential all the way back to a global minimum energy located at the Frank-Condon point of R CO ≈ 1.2 Å. This work, demonstrates that by forming polaritons, one can in principle manipulate photo-excited reactions via modification of the excited state pathways and curvatures. Fig. presents the first few polariton states generated from coupling a toluene molecule (under the cavity-free equilibrium nuclear geometry) to a single mode cavity. In particular, the first two electronic excited state states (orange) and one photon-dressed ground state (blue) are shown. At a weak coupling of λ = 0.01 a.u. and a cavity frequency that is in resonance with the first electronic excitation, the electronic and photonic DOF strongly mix and generate the polariton states (purple), resulting in the usual UP and LP polariton states. The second excited states, due to their offresonant frequency, is not explicitly involved into polariton formation under this particular coupling strength. For a larger coupling strength λ = 0.10 a.u. (Fig. ), the Rabi splitting Ω R (red arrow) becomes large enough to mix the UP with the second excited electronic state, thereby forcing the change in terminology to now include three polaritons: the LP, the upper lower polariton (ULP) and the upper upper polariton (UUP). In this case, the LP and ULP are strongly coupled through the light-matter interaction, while the ULP and UUP are interacting via the derivative couplings from the bare electronic interactions and DSE couplings mediated by the cavity. This will lead to interesting dynamical interplay between all DOFs; this was done explicitly in Ref. 193 using a simplified pQED Hamiltonian based on the data in Fig. to perform model polariton dynamics to elucidate the dynamical effects of a multi-level system.