## 3 Dipolar Contributions to the Mediated Rate
That is,  
\[<\Gamma^{(\mu m+m\mu)^{D}-(\mu\mu)^{A}}>=0. \tag{31}\]  
On taking advantage of the fact that the electric and magnetic dipole moments are both vectors, the isotropic rate between a magnetic dipole donor particle and an electric dipole-magnetic dipole acceptor species, and the rate between an electric dipole-magnetic dipole donor and a magnetic dipole acceptor, both mediated by an electric dipole polarizable third body, \(T\), also vanish on free orientational averaging, that is,  
\[<\Gamma^{(mm)^{D}-(\mu m+m\mu)^{A}}>=<\Gamma^{(\mu m+m\mu)^{D}-(mm)^{A}}>=0. \tag{32}\]  
We now return to Equation (21) and extract the contribution to the rate proportional to the product of electric and magnetic dipole moments at both the emitter and absorber sites. For oriented \(D\), \(A\), and \(T\), this is given by the expression  
\[ \begin{array}{l}\Gamma^{(\mu m+m\mu)^{D}-(\mu m+m\mu)^{A}}=\\ \frac{2m_{P_{f}}}{h}\{\mu_{i}^{0m}(D)m_{i^{\prime}}^{m0}(D)\}[\mu_{j^{0m}}^{ 0m}(A)m_{j^{\prime}}^{m0}(A)U_{ji}^{R^{\prime}}\overline{\
u}_{j^{\prime}l^{ \prime}}^{R^{\prime}}-\mu_{j^{0m}}^{0m}(A)m_{j^{\prime}}^{m0}(A)V_{ji}^{R^{ \prime}}\overline{U}_{j^{\prime}l^{\prime}}^{R^{\prime}}]V_{ik}^{R}\overline {U}_{l^{\prime}k^{\prime}}^{R}\\ +\mu_{j^{0m}}^{0m}(D)m_{i^{\prime}}^{m0}(D)[\mu_{j}^{0m}(A)m_{j^{\prime}}^{m0 }(A)V_{ji}^{R^{\prime}}\overline{U}_{j^{\prime}l^{\prime}}^{R^{\prime}}-\mu_{ j^{0m}}^{0m}(A)m_{j^{\prime}}^{m0}(A)U_{ji}^{R^{\prime}}\overline{V}_{j^{\prime}l^{ \prime}}^{R^{\prime}}]U_{ik}^{R}\overline{\
u}_{k^{\prime}l^{\prime}}^{R} \}\alpha_{ki}^{T}\overline{\alpha}_{k^{\prime}l^{\prime}}^{T},\end{array}  \tag{33}\]  
with explicit distance and orientational behavior following from substituting the geometric tensors. To find the isotropic rate, the orientational averages in Equations (28) and (29) are applied to Equation (33) and the product of tensors involving V and U are contracted. The transfer rate between two electric-magnetic dipole particles mediated by an electric dipole polarizable third molecule is found to be  
\[ \begin{array}{l}<\Gamma^{(\mu m+m\mu)^{D}-(\mu m+m\mu)^{A}}>=\\ -\frac{8\pi\rho_{f}}{2\pi\hbar c^{2}}\frac{1}{(4\pi\epsilon_{0})^{4}}\frac{k_{ \pi\mu}^{8}}{R^{2}R^{2}}(\hat{R}\cdot\hat{R}^{\prime})[\hat{\mu}^{0m}(D)\cdot \stackrel{{\rightarrow}}{{m}}^{m0}(D)][\hat{\mu}^{0m}(A)\cdot \stackrel{{\rightarrow}}{{m}}^{m0}(A)][\alpha_{\lambda\lambda}^{T} \overline{\alpha}_{\mu\mu}^{T}-\alpha_{\lambda\mu}^{T}\overline{\alpha}_{ \lambda\mu}^{T}].\end{array}  \tag{34}\]  
This rate is clearly discriminatory, dependent upon the chirality of the emitter and receiver, which manifests through the pseudoscalar quantity \([\hat{\mu}^{\lambda 0m}(\xi)\cdot\hat{m}^{m0}(\xi)]\), \(\xi=D\), \(A\), which changes sign when one enantiomer is replaced by its mirror-image structure. A number of other interesting features emerge from the result Equation (34). The rate vanishes if the acceptor is oriented perpendicularly relative to the donor and has its maximum value when \(D\), \(T\), and \(A\) are collinear. Similarly, if the product of the diagonal components of the polarizability tensor of the mediator is equal to its product of off-diagonal elements, the rate again vanishes. While this may be achieved in a two-level truncation scheme, a multi-level model of \(T\) is needed to ensure that the relay of energy occurs. Expression Equation (34) is retarded, as evidenced by the factor \(k_{m0}^{8}\), indicative of radiative migration of energy. Further confirmation is provided by the inverse square dependence of the rate on each of the inter-particle displacements, \(R\) and \(R^{\prime}\). Hence, no near-zone limit exists. This is to be expected since there is no coupling between static electric and magnetic dipoles. The result Equation (34) complements discriminatory pair RET between two optically active molecules [43; 44]. Dispersion energy shifts between two electric-magnetic dipole polarizable molecules are also discriminatory [54; 55; 56; 57; 58].