By far, there are two popular approaches in literature to solving this coupled electron-photon-nuclear system described in Eq. 108. Approach (I): solving the electron-nuclear problem followed by diagonalizing Eq. 55 with these electronically adiabatic basis states along with a photonic basis (e.g., number/Fock states, generalized coherent states, 148 polarized Fock States, 41 etc.). Approach (II): incorporating the photonic DOFs (through Eq. 55) into the common electronic structure framework whereby self-consistently solving the electron-photon-nuclear problem in one step. Both methods afford adiabatic polaritonic states as a result. This is because the two methods only only differ in the resulting basis describing the polaritonic system.