whose solution yields the time-dependent PEN wavefunction Ψ(r, q, R,t) and contains the complete information about the coupled system. Various concepts that are nowadays routinely used in the context of molecular dynamics have been generalized to PEN systems in cavities, described by the TDSE introduced above. An example is the Born-Oppenheimer approximation (BOA) , usually intended in the molecular framework to separate adiabatically the dynamics of the slow degrees freedom, typically the (heavy) nuclei, to the fast degrees of freedom, like the electrons and, sometimes, the protons. The BOA assumes that the faster electrons instantaneously adapt to the positions of the nuclei; consequently, if the electrons are initialized in an eigenstate, they remain in that eigenstate as long as the BOA remains valid. It follows that the electronic eigenenergy of the occupied state delineates the BO potential energy surface (PES) that governs the nuclear dynamics. Similarly, when dealing with a PEN system, the cavity-BOA has been introduced to separate the typical time-scales of the electronic degrees of freedom from those of the nuclear and photonic degrees of freedom. From the viewpoint of the quantized photon field, the cavity-BOA is physically justified by a small magnetic field since it is proportional to the momentum operator pν . In addition, interpreting somehow classically pν as related to the time derivative of the displacement qν , one can assume that in the cavity-BOA the changes in time of the electric field, which is proportional to qν , are slow. Then, electrons can adapt "quasi-instantaneously" to these slow changes in the electric displacement field. However, rapid changes of the electric field caused by strong interaction between light and matter may lead to the breakdown of the cavity-BO approximation.