## 4 Electric Dipole and Quadrupole Contributions to the Mediated Rate
 \tag{53}\]  
which exhibits \(R^{-6}\) behavior indicative of near-zone dipole-dipole transfer between \(D\) and \(T\), and \(R^{\prime-8}\) near-zone dipole-quadrupole exchange between \(T\) and \(A\). At the other distance extreme, approximating \(kR\) and \(kR^{\prime}\) to be much greater than unity gives rise to the far-zone asymptote  
\[\begin{array}{l}<\Gamma^{(\mu\mu)^{D}-(QQ)^{A}}>_{FZ}\sim\frac{\pi\rho_{f}}{450 h}\frac{1}{(4\pi\epsilon_{0})^{4}}\frac{k^{10}}{R^{2}{R^{\prime}}^{2}}|\overset{ \rightarrow}{\mu}^{0m}(D)|^{2}Q_{\rho\sigma}^{0m}(A)Q_{\rho\sigma}^{m0}(A)\{3 \bar{\alpha}_{\lambda\lambda}^{\rm T}\bar{\alpha}_{\mu\mu}^{T}+\alpha_{\lambda \mu}^{\rm T}\bar{\alpha}_{\lambda\mu}^{T}\}[1+(\hat{\[ \begin{split}&<\Gamma^{(QQ)^{D}-(QQ)^{A}}>=\frac{\pi\rho_{f}}{1500 \hbar}\frac{1}{(4\pi\omega_{0})^{4}}\frac{1}{R^{8}R^{8}}Q^{0m}_{v\tau\tau}(D)Q^{ 0m}_{v\tau}(A)Q^{m0}_{\rho\sigma}(A)\\ &\times\{\mathfrak{a}^{T}_{\lambda\lambda}\overline{\pi}^{T}_{ \mu\mu}[-1+3(\hat{R}\cdot\hat{R}^{\prime})^{2}](5184+2592k^{2}(R^{2}+{R^{ \prime}}^{2})+1080k^{4}(R^{4}+{R^{\prime}}^{4})+1296k^{4}{R^{2}}{R^{\prime}}^ {2}\\ &-72k^{6}(R^{6}+{R^{\prime}}^{6})+540k^{6}R^{2}{R^{\prime}}^{2}(R^ {2}+{R^{\prime}}^{2})+225k^{8}R^{4}{R^{\prime}}^{4}-36k^{8}R^{2}{R^{\prime}}^ {2}(R^{4}+R^{\prime})\\ &-15k^{10}R^{4}{R^{\prime}}^{4}(R^{2}+{R^{\prime}}^{2})+12k^{2} R^{6}{R^{\prime}}^{6})\\ &+{\mathfrak{a}^{T}_{\lambda\mu}\overline{\pi}^{T}_{\lambda\mu} (2592[41+2(\hat{R}\cdot\hat{R}^{\prime})^{2}]+2592[8+(\hat{R}\cdot\hat{R}^{ \prime})^{2}]k^{2}(R^{2}+{R^{\prime}}^{2})+1080[3+(\hat{R}\cdot\hat{R}^{\prime })^{2}]k^{4}(R^{4}+{R^{\prime}}^{4})\\ &+1296[3+(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{4}{R^{2}}{R^{ \prime}}^{2}+72[17-(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{6}(R^{6}+{R^{\prime }}^{6})+540[1+(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{6}{R^{2}}{R^{\prime}}^{2}(R ^{2}+{R^{\prime}}^{2})\\ &+45[1+5(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{8}{R^{4}}{R^{ \prime}}^{4}+36[7-(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{8}{R^{2}}{R^{\prime}}^ {2}(R^{4}+{R^{\prime}}^{4})+15[3-(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{10}{R^ {4}}{R^{\prime}}^{4}(R^{2}+{R^{\prime}}^{2})\\ &+[13+(\hat{R}\cdot\hat{R}^{\prime})^{2}]k^{12}{R^{6}}{R^{\prime }}^{6})\},\end{split}  \tag{56}\]  
which holds beyond the overlap of wave functions associated with \(D\), \(T\), and \(A\). Asymptotic limits of the general result Equation (56) readily follow from making the approximations appropriate to the near and far zones. At short \(R\) and \(R^{\prime}\),  
\[ \begin{split}&<\Gamma^{(QQ)^{D}-(QQ)^{A}}>_{NZ}\sim\frac{5184\pi \rho_{f}}{1500\hbar}\frac{1}{(4\pi\omega_{0})^{4}}\frac{1}{R^{8}R^{8}}Q^{0m}_{v \tau\tau}(D)Q^{m0}_{v\tau\tau}(D)Q^{0m}_{\rho\sigma}(A)Q^{m0}_{\rho\sigma}(A)\\ &\times\{[-1+3(\hat{R}\cdot\hat{R}^{\prime})^{2}]\mathfrak{a}^{T}_ {\lambda\lambda}\overline{\pi}^{T}_{\mu\mu}+\frac{1}{2}[41+2(\hat{R}\cdot\hat{ R}^{\prime})^{2}]\mathfrak{a}^{T}_{\lambda\mu}\overline{\pi}^{T}_{\lambda\mu} \},\end{split}  \tag{57}\]  
exhibiting inverse eighth power dependence on each of the inter-particle separation distances. In the far zone, on the other hand, radiative transfer leads to inverse square distance dependences,  
\[ \begin{split}&<\Gamma^{(QQ)^{D}-(QQ)^{A}}>_{FZ}\sim\frac{\pi\rho_{f} }{1500\hbar}\frac{1}{(4\pi\omega_{0})^{4}}\frac{k^{12}}{R^{2}R^{2}}Q^{0m}_{v\tau }(D)Q^{m0}_{v\tau\tau}(D)Q^{0m}_{\rho\sigma}(A)Q^{m0}_{\rho\sigma}(A)\\ &\times\{[-1+3(\hat{R}\cdot\hat{R}^{\prime})^{2}]\mathfrak{a}^{T}_ {\lambda\lambda}\overline{\pi}^{T}_{\mu\mu}+[13+(\hat{R}\cdot\hat{R}^{\prime}) ^{2}]\mathfrak{a}^{T}_{\lambda\mu}\overline{\pi}^{T}_{\lambda\mu}\}.\end{split}  \tag{58}\]