In the final example of the comparison between the scQED and the pQED schemes, an explicit test using either two (pink triangles) or three (purple squares) electronic states in the pQED approach compared to the variational scQED approach (blue circles), both at the Jaynes-Cummings level, was performed on the ethylene molecule, as shown in Fig. . Here, both the pQED as well as the scQED schemes used the generalized Jaynes-Cummings Hamiltonian (see Sec. 3.2). At low light-matter coupling strength λ < 0.05 a.u., the two-and three-electronic-state pQED-TD-DFT and the scQED-TD-DFT result in the same energies. At larger light-matter coupling strength λ > 0.05 a.u., none of the three methods are in agreement, indicating that more basis states are required to converge the pQED results. In this work, only the number of electronic states was explored, with only a single excited Fock state included in the basis. In this case, only the |0⟩ and |1⟩ states were used. In the supporting information of this work, the number of electronic states was further tested for large values of coupling. Here, the authors used up to 1000 electronic states, and neither the energies or dipoles were fully converged at this size of basis. There are a couple potential causes for this deviation. First, at this size of electronic basis, the number of included Fock states becomes extremely important for the convergence. For example, the |e j , n⟩ basis state could be very close in energy to some nearby |e k , n±1⟩ state, and recalling the block structure of Eq. 112, the interaction are then be non-zero if the transition dipole between electronic states |e j ⟩ and |e k ⟩ is non-zero, which is undeniably hard to predict for an arbitrary system (see examples in Fig. ). Additionally, the effects of the DSE terms that connect electronic basis states of the same photon number via the square of the dipole matrix (see Eq. 112) can mediate an interaction between a highenergy electronic state with a low-lying state, making the convergence of the number of electronic and Fock states of supreme importance. Careful convergence of the electronic and Fock states must be done carefully and simultaneously.