We suppose that CTMQC underestimates the initial increase in the photon number, and thus the population transfer, due to the complete neglect of the term ε geo in the TDPES and the corresponding contribution in the electronic evolution equation. This term contains second-order derivatives with respect to the photon-nuclear coordinate, which are expensive to compute numerically. In addition, it has been shown that it is indeed negligible when analyzing the EF factorization in terms of the (small) electron-nuclear mass ratio . In our PEN problem, this argument is invalid, and such term becomes important, or even dominant, as shown, for instance, in Fig. . The absence of this term prevents the approximated TDPES to develop a strong barrier around q = 0 in the Franck-Condon region, thus the distribution of trajectories cannot fully reproduce the QD density, with a shape that is reminiscent of the first excited state of the harmonic oscillator. This observation is supported by the results shown in Fig. , where CTMQC (blue line) -as well as MTE (red line) and TSH (orange line)perfectly agrees with QD results (black line) in the cavity-free case where the nonadiabatic dynamics of the electron-nuclear system is not affected by the strong coupling to the cavity. In Refs. [34,38], a similar behavior of the TDPES and of ε geo was observed, even though the factorization of the system was done in a different way. There, the PEN wavefunction was expressed as Ψ(r, q, R,t) = χ E (q,t)Φ E (r, R,t; q), and the ε geo showed a similar and significant contribution to the total TD-PES. This leads us to conclude that ε geo should be considered for further improvements in CTMQC to enhance the photon description.