Ref. 70 followed up on the work of Ref. 84, using an extended phase space path-integral framework, 414-416 so-called non-adiabatic ring-polymer molecular dynamics (NRPMD), to describe discrete electronic states using mapping variables and the photon field and nuclear DOF using the extend phase space variables of the ring polymer. Using the RPMD framework, the photon and nuclear degrees of freedom are copied into multiple 'beads' in the extended phase space, with the adjacent beads coupled through a Harmonic spring, forming a ring-polymer (shown schematically in Fig. ). This ring polymer, together with the electronic mapping variables, is evolved classically through the corresponding equation of motion. Despite the classical evolution, NRPMD effectively captures all possible quantum effects, including the electronic nonadiabatic effect and the nuclear quantum effects, as well as similar effects exhibited by the photonic DOF q c . Ref. 70 shows that even for a photon frequency as high as ω c = 500 meV, the rate constant predicted by the di- rect NRPMD simulations provides the same result as the rates obtained from the analytic MLJ theory in Eq. 205 (that uses the quantum description of a cavity photon mode), which matches the analytic result perfectly for the model calculation presented in panel (c). Recent work 420 has also used the adiabatic limit of RPMD (or referred to as path-integral MD) description of photon mode to perform molecular dynamics simulation using 'real' molecular systems (using classical force fields) beyond any simple model systems. Regarding such development of semiclassical methods for accurately capturing such cavity-modified reactivities, Ref. 421 introduced a Linearized Semi-classical approximation with Fermi's golden rule (FGR) rate theory, which can achieve the cavity-induced rate enhancement of such non-adiabatic electron transfer reactions with high accuracy.