## 2 QED Matrix Element for Third Body-Mediated RET between Two Chiral Molecules  
Consider RET between two chiral molecules, one serving as the donor species, \(D\), located at \(\overrightarrow{R}_{D}\), and the second, \(A\), positioned at \(\overrightarrow{R}_{A}\), playing the role of acceptor entity. If \(D\) and \(A\) are not identical, then their energy spectra need to overlap. Because symmetry plays a special part in classifying whether a molecule is chiral or not, spectroscopic selection rules are less stringent for such species relative to non-chiral or achiral substances, allowing multipole moment contributions other than the electric dipole moment to contribute to numerous processes involving the coupling of light with matter. \(D\) is assumed to initially be in the excited electronic state \(\mid\)m\({}^{\text{D}}\)\(>\), with energy \(E_{m}^{D}\), while \(A\) is in the ground state \(\mid\)0\({}^{\text{A}}\)\(>\), with energy \(E_{0}^{A}\). An amount of energy \(E_{m0}=E_{m}-E_{0}=hck_{m0}\) is transferred, with \(k_{m0}\) being the wave number of the transition associated with de-excitation in \(D\). This quantity of energy is gained by the absorber, which becomes excited to state \(\mid\)m\({}^{\text{A}}\)\(>\), while the emitter decays to the ground state, \(\mid\)0\({}^{\text{D}}\)\(>\). Let the exchanged energy be mediated by a third particle, \(T\), positioned at \(\overrightarrow{R}_{T}\), which serves to relay excitation between \(D\) and \(A\). \(T\) is electrically polarizable and may undergo virtual transitions from the ground state \(\mid\)0\({}^{\text{T}}\)\(>\) to excited levels \(\mid\)r\({}^{\text{T}}\)\(>\), but is otherwise inert, remaining in the ground state when not interacting. In the multipolar formalism of molecular QED theory [3; 4; 6], the total Hamiltonian operator for the system comprising the three particles, the radiation field, and their mutual interaction, is written as  
\[H=\sum_{\xi}H_{mol}(\xi)+H_{rad}+\sum_{\xi}H_{int}(\xi),\ \xi=D,A,T. \tag{1}\]  
The first term in Equation (1) is the molecular Hamiltonian of particle \(\xi\), \(H_{mol}(\xi)\), familiar from quantum chemistry and which is expressed as a sum of kinetic and intra-molecular potential energy contributions for a collection of non-relativistic charged particles grouped into atoms or molecules \(\xi\). A characteristic feature of QED is the placing of the electromagnetic field on an equal footing with matter, with both subject to the axioms of quantum mechanics. This leads to the explicit appearance in Equation (1) of the Hamiltonian operator for the radiation field, \(H_{rad}\). One form for the total energy of the field is given by  
\[H_{rad}=\frac{1}{2\epsilon_{0}}\int\{\overset{\rightarrow}{d}^{\perp,2}( \overrightarrow{r})+\epsilon_{0}^{2}c^{2}\overset{\rightarrow}{b}( \overrightarrow{r})\}d^{3}\overrightarrow{r}, \tag{2}\]  
where \(\overset{\rightarrow}{d}^{\perp}(\overrightarrow{r})\) and \(\overset{\rightarrow}{b}(\overrightarrow{r})\) are microscopic transverse electric displacement and magnetic fields, respectively. In QED, the electromagnetic field is pictured as a sum of independent simple harmonic oscillators, whose quantization is straightforward in either the particle or wave viewpoints. This results in photons, which are completely characterized by their individual mode designation \(\overset{\rightarrow}{k},\lambda\), where \(\overset{\rightarrow}{k}\) is the wave vector, signifying the propagation direction of light, and \(\lambda\) is its index of polarization. These first two terms of Equation (1) constitute the unperturbed Hamiltonian, whose solutions are assumed to be known. They correspond to the problem of matter in the absence of radiation, and the propagation of electromagnetic radiation in free space, respectively.