Until now, the nuclear gradient expressions were formulated in the adiabatic-Fock basis (i.e., photon-dressed electronic states), which is not the eigenbasis of V (defined in Eq. 12). Some MQC methods, such as FSSH, are formulated specifically in the eigenstates of V and require the calculation of the nuclear gradients on the eigenstates of V . For the molecule-cavity hybrid system, we define the polaritonic state as the eigenstate of V = Ĥpl (see definition in Eq. 12) through the following eigenequation