## 3 Dipolar Contributions to the Mediated Rate  
For weak radiation-matter coupling, in which a perturbative approximation is invoked to solve the QED eigenvalue problem via the computation of the matrix element, \(M_{fi}\), the rate of RET may be obtained from the Fermi golden rule formula:  
\[\Gamma=\frac{2\pi\rho_{f}}{h}|M_{fi}|^{2}, \tag{25}\]  
where \(\Gamma\) is the rate, and \(\rho_{f}\) is the density of final states. To keep the length of rate results manageable, specific multipole moment-dependent contributions will be considered separately. The first term to be examined is the rate between an electric dipole donor and an acceptor that is characterized by terms proportional to the product of transition electric and magnetic dipole moments, i.e., \"\(\mu\stackrel{{\rightarrow}}{{m}}+\mu\stackrel{{ \rightarrow}}{{m}}+\mu\stackrel{{\rightarrow}}{{m}}+\mu\stackrel{{ \rightarrow}}{{m}}\)\". The rate for the three species in fixed orientation relative to one another is  
\[ \begin{array}{ll}\Gamma^{(\mu\mu)^{D}-(\mu m+m\mu)^{A}}&=\frac{2\pi\rho_{f}}{h} \mu_{i}^{0m}(D)\overline{\mu}_{f^{\prime}}^{m0}(D)\\ &\times[\mu_{j}^{0m}(A)m_{j}^{m0}(A)V_{jl}^{R^{\prime}}\overline{U}_{jl^{ \prime}}^{R^{\prime}}-\mu_{j}^{0m}(A)m_{j}^{m0}(A)U_{jl}^{R^{\prime}} \overline{V}_{jl^{\prime}}^{R^{\prime}}]\\ &\times V_{jk}^{R}\overline{V}_{jk^{\prime}}^{R^{\prime}}\alpha_{kl}^{T} \overline{x}_{kl^{\prime}}^{T},\end{array}  \tag{26}\]where the overbar denotes the complex conjugate quantity, and use has been made of the relations \(m_{i}^{0m}=-m_{i}^{m0}=\overline{m}_{i}^{m0}\), for an imaginary magnetic dipole moment operator and real molecular wave functions. Explicit distance and orientational dependences of Equation (26) are obtained on inserting the geometric tensors \(V_{ij}^{R/R^{\prime}}\) and \(U_{ij}^{R/R^{\prime}}\). To obtain the rate formula applicable to species that are freely tumbling, as in the gaseous or liquid phases, an orientational average must be performed on Equation (26). The averages over \(D\) and \(A\) involve using the result for the average of a second-rank Cartesian tensor since they each involve a product of dipole moments. Hence  
\[<\mu_{i}^{0m}(D)\overline{m}_{i^{\prime}}^{m0}(D)>=\frac{1}{3}\delta_{ii^{ \prime}}\overset{\rightarrow}{\mu}^{m0}(D)|^{2}, \tag{27}\]  
and  
\[<\mu_{j}^{0m}(A)m_{j^{\prime}}^{m0}(A)>=\frac{1}{3}\delta_{ij^{\prime}}[ \overset{\rightarrow}{\mu}^{0m}(A)\cdot\overset{\rightarrow}{m}^{m0}(A)], \tag{28}\]  
where the chevron brackets indicate an orientationally averaged quantity. From Equation (26), it is seen that the mediator appears in the rate as the product of its dynamic polarizability and its complex conjugate, each of which are second-rank tensors. A free orientational average of the mediator thus requires use of the result for the average of a fourth-rank Cartesian tensor [53],  
\[<\alpha_{kl}^{T}\overline{\alpha}_{k^{\prime}l^{\prime}}^{T}>=\frac{1}{30}\{(4 \delta_{kl}\delta_{k^{\prime}l^{\prime}}-\delta_{kl^{\prime}}\delta_{k^{ \prime}l}-\delta_{kk^{\prime}}\delta_{ll^{\prime}})\alpha_{\lambda\lambda}^{T} \overline{\alpha}_{k^{\prime}\mu}^{T}+[-2\delta_{kl}\delta_{k^{\prime}l^{ \prime}}+3(\delta_{k^{\prime}l^{\prime}}\delta_{k^{\prime}l}+\delta_{kk^{ \prime}}\delta_{ll^{\prime}})]\alpha_{\lambda\mu}^{T}\overline{\alpha}_{ \lambda\mu}^{T}\}, \tag{29}\]  
using the index symmetry properties of the polarizability tensor, with Greek subscripts referring to Cartesian tensor components in the body-fixed frame. The result Equation (29) will be utilized in all of the remaining calculations of isotropic transfer rates. Inserting the last three relations in Equation (26) and contracting with the product of V and U tensors results in the isotropic rate vanishing, that is,  
\[<\Gamma^{(\mu\mu)^{D}-(\mu m+m\mu)^{A}}>=0. \tag{30}\]  
Unsurprisingly, interchanging the role of \(D\) and \(A\) also produces a vanishing randomly averaged rate.