Derivation of Eq. (113)

After inserting Eq. (111) into Eq. (112) we must sum up products of four Clebsh-Gordan coefficients, similarly as in Eq. (129), that is,

$$\sum_{{m_jm_sm'_jm'_s}} C^{{\textstyle{\frac{1}{2}}}m'_s}_{{\text... ...m_s} C^{j'm'_j}_{jm_jI''M''_I} C^{j'm'_j}_{l'm'_l{\textstyle{\frac{1}{2}}}m'_s}$$

$$= (-1)^{M''_I-m'_l-l-v'} \sqrt{2}\sqrt{2j'+1}\sqrt{2j'+1}\sqrt{2j+1}$$

$$\times \sum_{T'M'_T} C^{T'M'_T}_{l'm'_ll,-m_l} C^{T'M'_T}_{v',-\m... ...}{2}}} & {\textstyle{\frac{1}{2}}} & v' \\ l & l' & T' \end{array}\right\} ,$$ (130)

and then as in Eq. (131):

$$\sum_{m_lm'_lm_km'_k} (-1)^{-m'_l} C^{km_k}_{lm_lIM_I} C^{k'm'_k}_{l'm'_lI',-M'_I} C^{T'M'_T}_{l'm'_ll,-m_l} C^{k'm'_k}_{TM_Tkm_k}$$

$$= (-1)^{l+k+k'-I'}\sqrt{(2k+1)(2k'+1)(2k'+1)(2T'+1)}$$

$$\times \sum_{W'M'_W}(-1)^{M'_W-M'_I} C^{W'M'_W}_{T,-M_TT'M'_T} C^... ...rray}{rrr} l & k & I \\ l' & k' & I' \\ T' & T & W' \end{array}\right\} .$$ (131)

We can now perform the summation over $M'_I$ and $M_I$, which gives the factor $(-1)^{I+I'-L'}\delta_{L'W'}\delta_{M'_LM'_W}$ and allows for a summation over $W'$ and $M'_W$. After inserting all these results into Eq. (112), we obtain

$$\langle\phi_{N'l'j'}\vert\vert\tilde{h}^{I''}(\tilde{\rho}^{J''M''})\vert\vert\phi_{Nlj}\rangle \!\!\!\! = \sum_{M''_I} \sum_{n'L'v'J'}\sum_{mIm'I'} \sum_{M'_L\mu'} \int r^2{\rm d}\,{r}$$

$$\sum_{M'}C^{J'M'}_{L'M'_Lv'\mu'} {\textstyle{\frac{(-1)^{J'-M'}}{\sqrt{2J'+1}}}}(-1)^{m'} K^{n'L'}_{mIm'I'} (-1)^{m'+I'}$$

$$\times \sum_{TM_T} \sum_{k'} \sum_{k} {\textstyle{\frac{1}{\sqrt{2}}}} \... ...ac{1}{2}}}\rangle \sqrt{{\textstyle{\frac{(2T+1)(2k+1)}{4\pi}}}} C^{k'0}_{T0k0}$$

$$\times b^{3+n'} F^{k'*}_{m'I'N'l'}(br) \tilde{U}_{n'L'v'J'}^{TJ''}(br)F^{k}_{mINl}(br) e^{-2(br)^2}$$

$$\times C^{TM_T}_{J',-M'J''M''}$$

$$\times (-1)^{M''_I-l-v'} \sqrt{2}\sqrt{2j'+1}\sqrt{2j+1}$$

$$\times \sum_{T'M'_T} C^{T'M'_T}_{v',-\mu' I''M''_I} \left\{\begin... ...{1}{2}}} & {\textstyle{\frac{1}{2}}} & v' \\ l & l' & T' \end{array}\right\}$$

$$\times (-1)^{M'_L}(-1)^{l+k+k'-I'}\sqrt{(2T'+1)}$$

$$\times C^{L'M'_L}_{T,-M_TT'M'_T} \left\{\begin{array}{rrr} l & k & I \\ l' & k' & I' \\ T' & T & L' \end{array}\right\} (-1)^{I+I'-L'},$$ (132)

Now, we have to sum up products of three Clebsh-Gordan coefficients, similarly as in Eq. (133), that is,

$$\sum_{M'_LM'_T\mu'} (-1)^{M'_L} C^{J'M'}_{L'M'_Lv'\mu'} C^{T'M'_T}_{v',-\mu' I''M''_I} C^{L'M'_L}_{T,-M_TT'M'_T}$$

$$= {\textstyle{\sqrt{\frac{2T'+1}{2I''+1}}}}(-1)^{T'}{\textstyle{\sqrt{\frac{2L'+1}{2T+1}}}} (-1)^{L'+T'-T+M_T}$$

$$\times (-1)^{T'+L'+I''+J'}\sqrt{2I''+1}\sqrt{2J'+1} C^{T,-M_T}_{I... ...\left\{\begin{array}{rrr} v' & T' & I'' \\ T & J' & L' \end{array}\right\} .$$ (133)

This gives:

$$\langle\phi_{N'l'j'}\vert\vert\tilde{h}^{I''}(\tilde{\rho}^{J''M''})\vert\vert\phi_{Nlj}\rangle$$

$$= \sum_{M''_I} \sum_{n'L'v'J'}\sum_{mIm'I'} \sum_{M'} (-1)^{J'} K^{n'L'}_{mIm'I'} (-1)^{I'}$$

$$\times \sum_{TM_T} \sum_{k'} \sum_{k} {\textstyle{\frac{1}{\sqrt{2}}}} \... ...le{\frac{1}{2}}}\rangle \sqrt{{\textstyle{\frac{(2k+1)}{4\pi}}}} C^{k'0}_{T0k0}$$

$$\times \int r^2{\rm d}\,{r}\, b^{3+n'} F^{k'*}_{m'I'N'l'}(br) \tilde{U}_{n'L'v'J'}^{TJ''}(br)F^{k}_{mINl}(br) e^{-2(br)^2}$$

$$\times C^{TM_T}_{J',-M'J''M''} C^{T,-M_T}_{I'',-M''_IJ'M'}$$

$$\times (-1)^{I''+J'}(-1)^{T'} (-1)^{T}(-1)^{k+k'-I'-v'}(-1)^{I+I'-L'}$$

$$\times \sqrt{2}\sqrt{2j'+1}\sqrt{2j+1}\sqrt{(2T'+1)}\sqrt{(2L'+1)(2T'+1)}$$

$$\times \sum_{T'} \left\{\begin{array}{rrr} j & j' & I'' \\ {\textstyl... ...\left\{\begin{array}{rrr} v' & T' & I'' \\ T & J' & L' \end{array}\right\} .$$ (134)

Finally, summations over $M'$ and $M_T$ give the factor $\delta_{I''J''}\delta_{M''_IM''}$, that is:

$$\langle\phi_{N'l'j'}\vert\vert\tilde{h}^{I''}(\tilde{\rho}^{J''M''})\vert\vert\phi_{Nlj}\rangle$$

$$= \delta_{I''J''} (-1)^{J''} \sum_{n'L'v'J'}\sum_{mIm'I'} K^{n'L'... ...'}\vert\vert{\textstyle{\frac{1}{2}}}\rangle{\textstyle{\sqrt{\frac{1}{4\pi}}}}$$

$$\times \sum_{Tkk'} (-1)^{T} (2T+1) \int r^2{\rm d}\,{r}\, b^{3+n... ...k'}_{m'I'N'l'}(br) \tilde{U}_{n'L'v'J'}^{TJ''}(br)F^{k}_{mINl}(br) e^{-2(br)^2}$$

$$\times (-1)^{k} (-1)^{T}(-1)^{I-L'} C^{T0}_{k0k'0}{\textstyle{\sqrt{\frac{(2k+1)(2k'+1)(2L'+1)(2j+1)(2j'+1)}{(2T+1)}}}}$$

$$\times \sum_{T'} (-1)^{T'}(2T'+1) \left\{\begin{array}{rrr} j & j... ...\left\{\begin{array}{rrr} v' & T' & J'' \\ T & J' & L' \end{array}\right\} ,$$ (135)

where we have also used Eq. (90). This shows explicitly that in the field calculated for the density matrix with multipolarity $J''$, only the multipole $J''$ appears.

References