In principle, both pQED and scQED will generate identical results under the complete basis limit. Compared to scQED, the pQED scheme is much simpler in the sense that it does not require additional redevelopment of electronic structure theory for the QED Hamiltonian as well as the simplicity that comes with a non-self-consistent solution. With the above available examples, one can see that if the light matter coupling strength is high and more electronic states are needed for a fully converged pQED calculation, then in principle, one needs to fully converge these excited electronic states first before doing the pQED simulation. One important consideration, of many, is the fulfillment of the TRK sum rule in Eq. 68, which is a fundamental requirement by exact quantum mechanics. However, due to the use of approximate electronic structure methods, the TRK sum rule becomes method-dependent. For example, TD-DFT satisfies this rule but TD-DFT in the Tamm-Dancoff approximation (TDA) does not (see Sec. 3.2.5). Thus the accuracy of the excited state dipole matrix elements (such as those shown in Fig. ) might violate this rule for some electronic structure methods and become less accurate for high-lying excited states, which are a necessary input into the pQED method at large coupling strengths and will eventually lead to a less accurate description of polariton states. On the other hand, one can start with a reasonably sized basis, construct excited configurations from a reference trial electronic-photonic wavefunction, and solve only the first few polaritonic states, as needed, directly by using scQED approach with high accuracy combined wither iterative diagonalization techniques. Despite the enormous recent progress in both scQED and pQED schemes, what is generally missing is a consistent and fair comparison of both approaches and assessment of the strengths/limitations of each method under different scenarios. These consistent benchmarks will be one of the urgent tasks for the electronic structure community.