Finally, Ref. 36 investigated the VSC effect using a model that couples radiation modes to the vibrational degrees of freedom in non-adiabatic electron transfer reaction. The authors in that work considered an ensemble of molecules placed inside an optical cavity (schematically shown in Fig. ) with quantized radiation described by a single cavity mode. Each molecule has a reactant (donor) |R J ⟩ and a product |P J ⟩ electronic state (with J as the index for the molecule) and they are coupled to a high-frequency molecular vibration such that the electron transfer rate (reactant to product) constant is computed using the MLJ theory. The cavity excitation is assumed to be coupled to the vibrational excitation but only on the product such that the light-matter coupling term read g J â † c âi |P J ⟩⟨P J | + h.c. where g c is the light-matter coupling strength. In other words, a product state with no vibrational excitation and one photon in the cavity â † c |P J ⟩⊗| 0⟩ (where â † c is the cavity photon creation operator) is coupled to the product state with a vibrational excitation and no photons in the cavity â † J |P J ⟩ ⊗ | 0⟩ where | 0⟩ represents the vacuum state of the cavity and the molecular vibrations. For N identical molecular vibrations, only one collective bright vibration, representing delocalized vibrational excitation over all molecules, can be shown to hybridize strongly to the cavity excitation, such that the light-matter coupling can be written as √ N gâ † c âB + h.c. where â † B = 1 √ N J â † J |P J ⟩⟨P J |. At the same time, N -1 dark vibrational excitations remain uncoupled from the cavity photon mode. The resulting hybrid vibro-polaritonic states are schematically illustrated in Fig. .