## 2 QED Matrix Element for Third Body-Mediated RET between Two Chiral Molecules
 \tag{14}\]  
Converting the sum over discrete virtual photon wave vectors to a continuous variable via the relation,  
\[\sum\limits_{\vec{p}}\rightarrow\frac{V}{\left(2\pi\right)^{3}}\int d^{3} \vec{p}, \tag{15}\]  
and performing the necessary angular averages, the wave vector sums may be evaluated to give  
\[\sum\limits_{p}\left(\frac{p}{2\epsilon_{0}V}\right)(\delta_{ij}-\hat{p}_{i} \hat{p}_{j})\{\frac{e^{i\vec{p}\cdot\vec{R}}}{p-k_{m0}}+\frac{e^{-i\vec{j}\cdot \vec{R}}}{p+k_{m0}}\}=-V_{ij}(k_{m0},\stackrel{{\rightarrow}}{{R}}), \tag{16}\]  
and  
\[\frac{1}{c}\sum\limits_{\vec{p}}\left(\frac{p}{2\epsilon_{0}V}\right)\epsilon_ {ijk}\hat{p}_{k}\{\frac{e^{i\vec{j}\cdot\vec{R}}}{p-k_{m0}}+\frac{e^{-i\vec{j} \cdot\vec{R}}}{p+k_{m0}}\}=-U_{ij}(k_{m0},\stackrel{{\rightarrow} }{{R}}). \tag{17}\]  
On the right-hand side of the result Equation (16) is the familiar retarded resonant electric dipole-dipole interaction tensor, \(V_{ij}(k_{m0},\stackrel{{\rightarrow}}{{R}})\), defined by  
\[V_{ij}(k,\stackrel{{\rightarrow}}{{r}})=-\frac{1}{4\pi\epsilon_{0}}(- \
abla^{2}\delta_{ij}+\
abla_{i}\
abla_{j})\frac{e^{ikr}}{r}=\frac{1}{4\pi \epsilon_{0}r^{3}}[(\delta_{ij}-3\hat{r}\hat{r}_{j})(1-ikr)-(\delta_{ij}-\hat{ r}\hat{r}_{j})k^{2}r^{2}]e^{ikr}. \tag{18}\]  
Appearing in Equation (17) is the tensor  
\[U_{ij}(k,\stackrel{{\rightarrow}}{{r}})=-\frac{ik}{4\pi\epsilon_{0} c}\epsilon_{ijk}\
abla_{k}\frac{e^{ikr}}{r}=\frac{1}{4\pi\epsilon_{0}cr^{3}}\epsilon_{ ijk}\hat{r}_{k}[ikr+k^{2}r^{2}]e^{ikr}, \tag{19}\]  
which features in RET between electric and magnetic dipole moments; it is entirely retarded, vanishing when \(k=0\)[3, 4, 43]. The contribution Equation (14) thus becomes  
\[[\mu_{i}^{0m}(D)m_{j}^{\prime m0}(A)V_{ik}(k_{m0},\stackrel{{ \rightarrow}}{{R}})U_{jl}(k_{m0},\stackrel{{\rightarrow}}{{R}}^{ \prime})+\mu_{i}^{0m}(D)\mu_{j}^{\prime m0}(A)U_{ik}(k_{m0},\stackrel{{ \rightarrow}}{{R}})V_{jl}(k_{m0},\stackrel{{\rightarrow}}{{R}}^{ \prime})]\alpha_{kl}(T;-k_{m0},+k_{m0}). \tag{20}\]Returning to Equation (10) and focusing on the pure and mixed dipolar terms only, using the identities Equations (11) and (12), and the results Equations (16) and (17), the electric and magnetic dipole-dependent matrix element is found to be  
\[ \begin{array}{ll}M_{fi}^{\mu+m}&=[-\mu_{i}^{0 mn}(D)\mu_{j}^{m0}(A)V_{ik}^{R}V_ {jl}^{R^{\prime}}+\mu_{i}^{0m}(D)\eta_{j}^{m0}(A)V_{ik}^{R}U_{jl}^{R^{\prime}} \\ &+m_{i}^{0m}(D)\mu_{j}^{m0}(A)U_{ik}^{R}V_{jl}^{R^{\prime}}-m_{i}^{0m}(D)m_{j}^ {m0}(A)U_{ik}^{R}U_{jl}^{R^{\prime}}|\alpha_{kl}^{T},\end{array}  \tag{21}\]  
where to simplify the expression, the following shorthand notation is introduced: \(V_{ij}^{R}=V_{ij}(k_{m0},\stackrel{{\rightarrow}}{{R}})\), \(U_{ij}^{R}=U_{ij}(k_{m0},\stackrel{{\rightarrow}}{{R}})\), and \(\alpha_{kl}^{T}=\alpha_{kl}(T;-k_{m0},+k_{m0})\). A similar procedure is used to obtain contributions dependent upon the electric quadrupole moment. Only pure and mixed electric multipole moment terms of Equation (10) are extracted, with higher-order mixed magnetic dipole-electric quadrupole factors being neglected. Use is made of the result  
\[ \begin{array}{ll}\sum\limits_{\stackrel{{\rightarrow}}{{p}}} \biggl{(}\frac{p}{2\epsilon_{0}V}\biggr{)}(\delta_{ij}-\hat{p}_{i}\hat{p}_{j}) \
abla_{k}\{\frac{e^{i\vec{P}\cdot\vec{R}}}{p+k_{m0}}+\frac{e^{-i\vec{P}\cdot \vec{R}}}{p-k_{m0}}\}=-V_{ijk}(k_{m0},\stackrel{{\rightarrow}}{{R}}),\end{array}  \tag{22}\]  
which features in quadrupolar RET [52], with \(V_{ijk}(k,\stackrel{{\rightarrow}}{{r}})\) defined as  
\[ \begin{array}{ll}V_{ijk}(k,\stackrel{{\rightarrow}}{{r}})=- \frac{1}{4\pi\epsilon_{0}}(-\
abla^{2}\delta_{ij}+\
abla_{i}\
abla_{j}) \
abla_{k}\frac{e^{i\
u}}{r}\\ =\frac{1}{4\pi\epsilon_{0}r^{4}}[(\delta_{ij}-\hat{r}_{i}\hat{r}_{j})\hat{r}_ {k}(-ik^{3}r^{3}+k^{2}r^{2})+(\delta_{ij}\hat{r}_{k}+\delta_{ik}\hat{r}_{j}+ \delta_{jk}\hat{r}_{i}-5\hat{r}_{i}\hat{r}_{j}\hat{r}_{k})(k^{2}r^{2}+3 ikr-3)] e^{i\
u r}.\end{array}