However, if the molecular system contains a quantum degree of freedom (such as a vibration with frequency ℏω v β ≫ 1), the rate constant requires a quantum description beyond the simple Marcus theory (especially in the inverted regime). This is because, with quantum degrees of freedom, the system can also access vibrational excited states for which the driving force is no longer just ∆G but is modified by nℏω v , where n is the quantum number of vibrations and ω v is the vibrational frequency. The above situation is precisely the case when a non-adiabatic electron transfer reaction inside an optical cavity is considered. Such a setup is schematically shown in Fig. for plasmonic cavity. Note that the cavity frequency is either in the infrared regime or in the UV-Vis region, but not matching any particular vibrational transition or electronic transitions. Instead, the cavity mode is directly coupled to the transition dipole of the charge transfer process 84 between the |D⟩ and |A⟩ state, which is µ DA . Due to the presence of the cavity photon mode, new photon-dressed donor and acceptor states, such as |D⟩ ⊗ |n⟩ (donor state with n photons in the cavity) and |A⟩ ⊗ |n⟩ (acceptor state with n photons in the cavity), become available for mediating the charge transfer process. The potential energy surface for these states along the charge transfer reaction coordinate (a collective solvent coordinate, which is often referred to as the Marcus coordinate) is shown in Fig. . The polaritonmediated electron transfer rate constant can then be computed by considering all possible reactive channels |D⟩ ⊗ |n⟩ → |A⟩ ⊗ |m⟩. The chemical rate constant in the presence of quantum degrees of freedom can be computed using the Marcus-Levich-Jortner (MLJ) theory as follows . In addition to this, suppression of the chemical reactivity is also observed around the peak of the rate curve. This suppression is due to the presence of the permanent dipoles 84 that reduces the diabatic coupling between |D, 0⟩ and |A, 0⟩ by the factor S m,n as V DA • S nm = V DA exp -1 2 ((µ AA -µ DD )A 0 /ℏ) 2 , a result that is obtained by performing the polaron transformation, but can also be understood through the polarized Fock state formalism (Sec. 3.1.2). Overall, this work 84 points out that the cavity photon mode can act like high-frequency quantum vibration that modifies the non-adiabatic electron transfer rate constant, especially for driving forces in the inverted regime.