where |Ψ(t ) ⟩ is the total quantum state of the electronic-nuclear-photonic hybrid system. The time-evolution of such a system is governed by the Pauli-Fierz QED Hamiltonian ĤPF (Eq. 13). Depending on this complexity of the molecular system, one may perform the dynamics exactly as dictated by the TDSE (which may be prohibitively expensive for more than a few nuclear DOFs) or resort to various approximations, such as mixed quantum-classical (MQC) ap- proaches, semi-classical approaches, various approximate master equation approaches (e.g., Lindblad, Redfield, etc.), or approximate wavefunction approaches (e.g., multi-configurational Hartree-Fock). In the following discussion, we will briefly introduce an exact method for solving polariton quantum dynamics (Sec. 4.1) as well as a few popular mixed quantum-classical approaches (Sec. 4.2). Additionally, we will address the calculation of the exact nuclear gradients and their effects on the quantum dynamics (Sec.4.3).