## 4 Electric Dipole and Quadrupole Contributions to the Mediated Rate  
Pure electric dipole and electric quadrupole contributions to the third body-mediated rate, as well as various mixed contributions, may be obtained from the matrix element, \(M_{fi}^{\mu+Q}\), Equation (24). Individual terms are again considered separately. The first contribution to be studied is the rate term proportional to a donor that is an electric dipole emitter, and a mixed electric dipole-quadrupole acceptor, namely between \"\((\mu\mu)^{D}\)\" and \"\((\mu Q+Q\mu)^{A}\)\". Hence, for the three bodies in fixed mutual orientation,  
\[ \begin{split}&\Gamma^{(\mu\mu)^{D}-(\mu Q+Q\mu)^{A}}=\frac{2\pi \rho_{f}}{h}[\mu_{i}^{0m}(D)\mu_{f}^{0m}(D)\mu_{k}^{m0}(A)Q_{k^{\prime}l^{ \prime}}^{m0}(A)V_{k\
u}^{R}\overline{V}_{i^{\prime}r^{\prime}}^{R}V_{ks}^{R^{ \prime}}\overline{V}_{k^{\prime}s^{\prime}l^{\prime}}^{R^{\prime}}\\ &+\mu_{i}^{0m}(D)\mu_{i^{\prime}}^{0m}(D)Q_{kl}^{m0}(A)\mu_{k^{ \prime}}^{m0}(A)V_{kl}^{R}\overline{V}_{k^{\prime}r^{\prime}}^{R^{\prime}}V_{ks }^{R^{\prime}}|x_{rs}^{T}\overline{x}_{r^{\prime}s^{\prime}}^{T},\end{split}  \tag{44}\]  
with explicit distance and orientation dependences obtained on inserting the respective geometric tensors from Equations (18) and (23). To obtain the isotropic rate, the averages over \(D\) and \(T\) may be performed using the results Equations (27) and (29), respectively. For \(A\), however, an average over a third-rank Cartesian tensor is needed. This is given by  
\[<\mu_{i}^{0m}(\xi)Q_{jk}^{m0}(\xi)>=\frac{1}{6}\epsilon_{ijk}\epsilon_{\lambda \mu\
u}\mu_{\lambda}^{0m}(\xi)Q_{\mu\
u}^{m0}(\xi), \tag{45}\]  
which vanishes because the electric quadrupole moment is symmetric in the indices \(\mu\) and \(\
u\), while the Levi-Civita tensor is anti-symmetric in this pair. Thus  
\[<\Gamma^{(\mu\mu)^{D}-(\mu Q+Q\mu)^{A}}>=0. \tag{46}\]  
Based on symmetry considerations, the isotropic rate also vanishes when the characteristics of \(D\) and \(A\) are switched relative to the previous case, that is,  
\[<\Gamma^{(\mu Q+Q\mu)^{D}-(\mu\mu)^{A}}>=0. \tag{47}\]  
Using index symmetry of the electric quadrupole moment and index anti-symmetry of the alternating tensor, it is easily seen that the following isotropic electric quadrupole-dependent contributions to the rate are also zero,  
\[<\Gamma^{(\mu Q+Q\mu)^{D}-(QQ)^{A}}>=<\Gamma^{(QQ)^{D}-(\mu Q+Q\mu)^{A}}>=< \Gamma^{(\mu Q+Q\mu)^{D}-(\mu Q+Q\mu)^{A}}>=0. \tag{48}\]  
The last of these three written in Equation (48) is the analog of the discriminatory rates Equations (33) and (34) evaluated in the previous section involving the interference of electric and magnetic dipole coupling terms. While the electric dipole-quadrupole counterpart does not survive orientational averaging, the rate for fixed mutual orientation of the three bodies does depend on the chirality of \(D\) and \(A\). From the matrix element Equation (24), the rate is  
\[ \begin{split}&\Gamma^{(\mu Q+Q\mu)^{D}-(\mu Q+Q\mu)^{A}}=\\ &-\frac{2\pi\rho_{f}}{h}\{\mu_{i}^{0m}(D)Q_{i^{\prime}j^{\prime}} ^{0m}(D)[\mu_{k}^{m0}(A)Q_{k^{\prime}l^{\prime}}^{m0}(A)V_{ks}^{R^{\prime}} \overline{V}_{k^{\prime}s^{\prime}l^{\prime}}^{R^{\prime}}+Q_{kl}^{m0}(A)\mu _{k^{\prime}}^{m0}(A)V_{kl}^{R^{\prime}}\overline{V}_{k^{\prime}s^{\prime}}^{ R^{\prime}}]V_{ir}^{R}\overline{V}_{i^{\prime}r^{\prime}l^{\prime}}^{R} \tag{49}\]  
\[+Q_{ij}^{0m}(D)\mu_{i^{\prime}}^{0m}(D)[Q_{kl}^{m0}(A)\mu_{k^{\prime}}^{m0}(A) V_{kl}^{R^{\prime}}\overline{V}_{k^{\prime}s^{\prime}}^{R^{\prime}}+\mu_{k}^{m0}(A)Q _{k^{\prime}l^{\prime}}^{m0}(A)V_{ks}^{R}\overline{V}_{k^{\prime}s^{\prime}l^ {\prime}}^{R^{\prime}}]V_{ir}^{R}\overline{V}_{i^{\prime}r^{\prime}}^{R}]\}a_{ rs}^{T}\overline{a}_{rs}^{T}\overline{a}_{rs}^{T}.\end{split} \]  
The only surviving quadrupolar-dependent contributions to the isotropic rate are of the type \"\((\mu\mu)^{D}-(QQ)^{A}\)\", \"\((QQ)^{D}-(\mu\mu)^{A}\)\", and the pure quadrupole coupling term at each \(D\) and \(A\) center, \"\((QQ)^{D}-(QQ)^{A}\)\". These are now evaluated.