For ab initio non-adiabatic dynamics of realistic molecules, the difficulty often is obtaining the necessary components for the propagation of the nuclear and electronic DOFs, such as the gradients of the PESs ∇ R E a and more non-trivially the derivative couplings between electronic states d αβ . In polaritonic systems, one encounters new terms which contain gradients on the electronic dipole operators through the light-matter coupling as well as the dipole self-energy terms in the PF Hamiltonian (Eq. 104), which can be understood as