## 2 QED Matrix Element for Third Body-Mediated RET between Two Chiral Molecules
A convenient form to express these microscopic Maxwell field operators is as a sum over Fourier modes, with expansions  
\[\stackrel{{\rightarrow\perp}}{{d}}(\overrightarrow{r})=i\sum_{ \overrightarrow{k},\lambda}\biggl{(}\frac{hck\epsilon_{0}}{2V}\biggr{)}^{1/2} \bigl{[}\stackrel{{\rightarrow}}{{e}}^{(\lambda)}( \stackrel{{\rightarrow}}{{k}})a^{(\lambda)}(\stackrel{{ \rightarrow}}{{k}})e^{i\overrightarrow{k}\cdot\overrightarrow{r}}- \overrightarrow{e}^{(\lambda)}(\stackrel{{\rightarrow}}{{k}})a^ {\dagger(\lambda)}(\stackrel{{\rightarrow}}{{k}})e^{-i \overrightarrow{k}\cdot\overrightarrow{r}}\bigr{]}, \tag{4}\]  
and  
\[\stackrel{{\rightarrow}}{{b}}(\overrightarrow{r})=i\sum_{ \overrightarrow{k},\lambda}\biggl{(}\frac{hk}{2\epsilon_{0}cV}\biggr{)}^{1/2} \bigl{[}\stackrel{{\rightarrow}}{{b}}^{(\lambda)}( \stackrel{{\rightarrow}}{{k}})a^{(\lambda)}(\stackrel{{ \rightarrow}}{{k}})e^{i\overrightarrow{k}\cdot\overrightarrow{r}}- \overrightarrow{b}^{(\lambda)}(\stackrel{{\rightarrow}}{{k}})a^ {\dagger(\lambda)}(\stackrel{{\rightarrow}}{{k}})e^{-i \overrightarrow{k}\cdot\overrightarrow{r}}\bigr{]}. \tag{5}\]  
Appearing in these last two expressions are the second quantized raising and lowering operators for a \(\stackrel{{\rightarrow}}{{k}}\), \(\lambda\) mode photon, \(a^{\dagger(\lambda)}(\stackrel{{\rightarrow}}{{k}})\) and \(a^{(\lambda)}(\stackrel{{\rightarrow}}{{k}})\), respectively, with \(\stackrel{{\rightarrow}}{{e}}^{(\lambda)}(\stackrel{{ \rightarrow}}{{k}})\) being the complex unit electric polarization vector, \(\stackrel{{\rightarrow}}{{b}}^{(\lambda)}(\stackrel{{ \rightarrow}}{{k}})\) is its magnetic counterpart, and \(V\) is the volume of the radiation quantization box, which serves to restrict the number of allowed field modes to a countable infinity. Because the third body, \(T\), acts as a bridge between \(D\) and \(A\), scattering virtual photons, it is advantageous to employ the following form for the interaction operator, \(H_{int}(T)\),  
\[H_{int}(T)=-\epsilon_{0}^{-2}\alpha_{ij}(T)d_{i}^{\perp}(\overrightarrow{R}_{ T})d_{j}^{\perp}(\overrightarrow{R}_{T}), \tag{6}\]  
which is seen to be quadratic in the electric displacement field. The mediator responds to radiation through its ground state frequency-dependent electric dipole polarizability tensor  
\[\alpha_{ij}(T)=\alpha_{ij}(T_{z}\pm k_{1},\pm k_{2})=\sum_{r}\{\frac{\mu_{i}^{ 0r}(T)\mu_{j}^{r0}(T)}{E_{r0}\mp hck_{1}}+\frac{\mu_{j}^{0r}(T)\mu_{i}^{r0}(T)} {E_{r0}\mp hck_{2}}\}, \tag{7}\]  
where the matrix element of the electric dipole moment operator between electronic states \(10^{7}\)> and \(1\,r^{7}\)> of \(T\) is abbreviated to \(\mu_{i}^{0r}(T)=<0^{7}|\mu_{i}(T)|r^{T}>\). Since a single virtual photon traverses between \(D\) and \(T\), and between \(T\) and \(A\), the interaction Hamiltonian Equation (6) is an effective two-photon perturbation operator [45]. Employing it will help to simplify the computation of the matrix element by reducing the number of Feynman-like diagrams that have to be drawn and summed over, which is beneficial when solving higher-order processes. This type of coupling Hamiltonian has been used to similar good effect in the evaluation of the Casimir-van der Waals dispersion potential, which is attributed to two virtual photon exchanges [46; 47; 48; 49], and its radiation-induced analog [50; 51]. To complete the statement of the problem, the final item required is the specification of the initial and final states. These may be written as  
\[|i>=|m^{D},0^{A},0^{T}>;\ |f>=|0^{D},m^{A},0^{T}>, \tag{8}\]corresponding to the migration of electronic energy from \(D\) to \(A\) resonantly, with energy \(E_{m0}\), with the mediator \(T\) remaining unexcited before and after the transfer process. No photons--real or virtual--are present at times \(t<t_{i}\) and \(t>t_{f}\). Let the single virtual photon exchanged between \(D\) and \(T\) be of mode \(\overset{\rightarrow}{p},\varepsilon\), and the one propagating between \(T\) and \(A\) be of mode \(\overset{\rightarrow}{p}^{\prime},\varepsilon^{\prime}\).