x j a are the coordinated of nucleus a in the direction x j . For additional details on using the KRR method for molecular properties in polaritonics, we refer the reader to Ref. 175. Fig. presents recent work 175 by Hu and Huo using the KRR approach to parameterize the transition and permanent dipoles of the azomethane molecule (Fig. ), which exhibits a conical intersection between the ground S 0 and excited S 1 electronic states as a function of the dihedral angle. Inside the cavity (Fig. ), the ground and excited states hybridize (black box) to form exciton-polaritons { E j } with mixed electronic (primarily S 0 and S 1 ) and well as photonic character (primarily n = 0 and 1 vacuum Fock states). To perform the quantum dynamics simulations, the molecular dipole and its gradient were trained based on the KRR approach to yield the analytic dipole matrix elements, shown in Fig. against the exact results using the CASSCF electronic structure method as the training/exact data. The total polaritonic gradient of the first R E 1 and second R E 2 excited polaritonic states is shown in Fig. (see Eq. 64). Using these learned values, the quantum dynamics simulation of the photoisomerization of azomethane results are shown in Fig. . Inside the cavity, at a small coupling strength of A 0 = 0.05 a.u., the avoided crossing between the first and second excited polaritonic states (black box in Fig. ) reduces the amount of product formed by the photo-reaction by bifurcating the wavepacket before reaching the conical intersection. At an increased coupling strength (A 0 = 0.50 a.u.), this effect is increased, reducing the overall population of the trans-product to less than 40% from the original product yield of 60% by increasing the population transfer at the polaritonic avoided crossing region.