## 4 Electric Dipole and Quadrupole Contributions to the Mediated Rate
For a pure electric dipole donor and a pure electric quadrupole absorber, the oriented rate term from the matrix element Equation (24) is  
\[\Gamma^{(\mu\mu)^{D}-(QQ)^{A}}=\frac{2\pi\rho_{f}}{h}\mu_{i}^{0m}(D)\mu_{i^{ \prime}}^{m0}(D)Q_{kl}^{m0}(A)Q_{k^{\prime}l^{\prime}}^{m0}(A)\alpha_{rs}^{T} \overline{a}_{rs}^{T}V_{ir}^{R}\overline{V}_{i^{\prime}r^{\prime}}^{R}V_{ks}^{R^ {\prime}}\overline{V}_{k^{\prime}s^{\prime}l^{\prime}}^{R^{\prime}}. \tag{50}\]To derive the rate applicable to freely tumbling molecules, the rotational averages Equations (27) and (29) are again used for \(D\) and \(T\), respectively; for the product of electric quadrupole moments of \(A\), the following fourth-rank Cartesian tensor average is employed:  
\[<Q_{kl}^{0m}(A)Q_{k^{\prime}l^{\prime}}^{m0}(A)>=\frac{1}{10}(\delta_{kl^{\prime }}\delta_{k^{\prime}l}+\delta_{kl^{\prime}}\delta_{ll^{\prime}})Q_{\rho\sigma}^{0 m}(A)Q_{\rho\sigma}^{m0}(A), \tag{51}\]  
on making use of the symmetric and traceless properties of the tensor \(Q_{ij}^{0m}(\xi)\). Substituting the results for the averages, contracting indices, and multiplying the geometric tensors using relations Equations (18) and (23) yields the isotropic rate formula  
\[ \begin{array}{l}<\Gamma^{(\mu\mu)^{D}-(QQ)^{A}}>=\frac{\pi\rho_{f}}{450h} \frac{1}{(4\pi\epsilon_{0})}\frac{1}{\kappa^{6}R^{\prime 8}}|\overset{ \rightarrow}{\mu}^{0m}(D)|^{2}Q_{\rho\sigma}^{0m}(A)Q_{\rho\sigma}^{m0}(A)\\ \times[\alpha_{\lambda\lambda}^{\rm T}\bar{\alpha}_{\mu\mu}^{T}\{-4[54-243( \hat{R}\cdot\hat{R}^{\prime})^{2}]-108[1-3(\hat{R}\cdot\hat{R}^{\prime})^{2}] k^{2}{R^{\prime}}^{2}-360[1-3(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{2}{R^{2}} \\ +72[1-3(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{4}{R^{4}}-[21+9(\hat{R}\cdot\hat{ R}^{\prime})^{2}]k^{4}{R^{4}}+18[1-45(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{4}{R^{2}} {R^{\prime}}^{2}\\ +3[1-3(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{6}{R^{\prime}}^{6}-3[9+5(\hat{R} \cdot\hat{R}^{\prime})^{2}]k^{6}{R^{2}}{R^{\prime}}^{4}-18[1+3(\hat{R}\cdot \hat{R}^{\prime})^{2}]k^{6}{R^{4}}{R^{\prime}}^{2}\\ +[13-15(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{8}{R^{2}}{R^{\prime}}^{6}-3[3-( \hat{R}\cdot\hat{R}^{\prime})^{2}]k^{8}{R^{4}}{R^{\prime}}^{4}+3[1+(\hat{R} \cdot\hat{R}^{\prime})^{2}]k^{10}{R^{4}}{R^{\prime}}^{6}\\ +\alpha_{T}^{T}\bar{\alpha}_{\lambda\lambda}^{\rm T}\{3528+324(\hat{R}\cdot \hat{R}^{\prime})^{2}+[684+108(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{2}{R^{ \prime}}^{2}+360[3+(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{2}{R^{2}}\\ +72[17-(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{4}{R^{4}}+[89-3(\hat{R}\cdot\hat {R}^{\prime})^{2}]k^{4}{R^{\prime}}^{4}+[198+270(\hat{R}\cdot\hat{R}^{\prime })^{2}]k^{4}{R^{2}}{R^{\prime}}^{2}\\ +[41-3(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{6}{R^{\prime}}^{6}+[23-5(\hat{R} \cdot\hat{R}^{\prime})^{2}]k^{6}{R^{2}}{R^{\prime}}^{4}+18[13-(\hat{R}\cdot \hat{R}^{\prime})^{2}]k^{6}{R^{4}}{R^{\prime}}^{2}\\ +5[3-(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{8}{R^{2}}{R^{\prime}}^{6}+29[1+( \hat{R}\cdot\hat{R}^{\prime})^{2}]k^{8}{R^{4}}{R^{\prime}}^{4}+[1+(\hat{R} \cdot\hat{R}^{\prime})^{2}]k^{10}{R^{4}}{R^{\prime}}^{6})],\end{array}  \tag{52}\]  
which holds for all \(R\) and \(R^{\prime}\) outside the charge overlap region, with \(k=k_{m0}\). A true near-zone limit of Equation (52) arises since static coupling is possible between electric multipoles. Retaining the pertinent wave vector independent terms, the isotropic rate valid at short separations between particles is  
\[ \begin{array}{l}<\Gamma^{(\mu\mu)^{D}-(QQ)^{A}}>_{NZ}\sim\frac{\pi\rho_{f}}{450h }\frac{1}{(4\pi\epsilon_{0})^{4}}\frac{1}{R^{6}R^{\prime}}^{1}\mu^{-0m}(D)|^{2 }Q_{\rho\sigma}^{0m}(A)Q_{\rho\sigma}^{m0}(A)\\ \times\{-4\alpha_{\lambda\lambda}^{\rm T}\bar{\alpha}_{\mu\mu}^{T}[54-243( \hat{R}\cdot\hat{R}^{\prime})^{2}]+\alpha_{\lambda\mu}^{\rm T}\bar{\alpha}_{ \lambda\mu}^{\rm T}\{3528+324(\hat{R}\cdot\hat{R}^{\prime})^{2}]\},\end{array}