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Next: Derivation of Eq. (113) Up: Solution of self-consistent equations Previous: Calculation of coefficients (72)


Derivation of Eq. (98)

We begin by a summation of the product of four Clebsh-Gordan coefficients, that appear in Eq. (97), that is,

    $\displaystyle \sum_{{mm_sm'm'_s}} C^{{\textstyle{\frac{1}{2}}}m'_s}_{{\textstyl... ...{1}{2}}}m_s} C^{jm}_{j'm'J''M''} C^{j'm'}_{l'm'_l{\textstyle{\frac{1}{2}}}m'_s}$  
  $\textstyle =$ $\displaystyle \sum_{{mm_sm'm'_s}} (-1)^{{\textstyle{\frac{1}{2}}}-m_s}{\textsty... ...xtstyle{\sqrt{\frac{2j+1}{2l+1}}}}C^{l,-m_l}_{j,-m{\textstyle{\frac{1}{2}}}m_s}$  
    $\displaystyle \times (-1)^{j'-m'}{\textstyle{\sqrt{\frac{2j+1}{2J''+1}}}}C^{J''... ...le{\sqrt{\frac{2j'+1}{2l'+1}}}}C^{l'm'_l}_{{\textstyle{\frac{1}{2}}},-m'_sj'm'}$  
  $\textstyle =$ $\displaystyle (-1)^{{\textstyle{\frac{1}{2}}}}(-1)^{{\textstyle{\frac{1}{2}}}}(... ...\textstyle{\sqrt{\frac{2j+1}{2J''+1}}}}{\textstyle{\sqrt{\frac{2j'+1}{2l'+1}}}}$  
    $\displaystyle \times (-1)^{1-v} (-1)^{j+{\textstyle{\frac{1}{2}}}-l}\sum_{{mm_s... ...{{\textstyle{\frac{1}{2}}}m'_sj'm'} C^{l,-m_l}_{{\textstyle{\frac{1}{2}}}m_sjm}$  
  $\textstyle =$ $\displaystyle (-1)^{j'-m'_l+j-l+1-v} \sqrt{2}\sqrt{2j+1}\sqrt{2j+1}\sqrt{2j'+1}$  
    $\displaystyle \times \sum_{T'M'_T} C^{T'M'_T}_{l'm'_ll,-m_l} C^{T'M'_T}_{v,-\mu... ...1}{2}}} & {\textstyle{\frac{1}{2}}} & v \\ l & l' & T' \end{array}\right\} ,$ (123)

see Eq. 8.7(20) in Ref. [6].

The product of spherical harmonics reads

$\displaystyle Y_{km_k}(\theta,\phi)Y^*_{k'm'_k}(\theta,\phi)$ $\textstyle =$ $\displaystyle (-1)^{m'_k} \sum_{TM_T} {\textstyle{\sqrt{\frac{(2k+1)(2k'+1)}{4\pi(2T+1)}}}}$  
    $\displaystyle \times C^{T0}_{k0k'0}C^{TM_T}_{km_kk',-m'_k} Y_{TM_T}(\theta,\phi) ,$ (124)

see Eqs. 5.4(1) and 5.6(9) in Ref. [6]. This gives us another sum of products of four Clebsh-Gordan coefficients to sum up:
    $\displaystyle \sum_{m_lm'_lm_km'_k} (-1)^{m'_k-m'_l} C^{km_k}_{lm_lUM_U} C^{k'm'_k}_{l'm'_lU',-M'_U} C^{T'M'_T}_{l'm'_ll,-m_l} C^{TM_T}_{km_kk',-m'_k}$  
  $\textstyle =$ $\displaystyle \sum_{m_lm'_lm_km'_k} (-1)^{m'_k-m'_l} (-1)^{l-m_l}{\textstyle{\sqrt{\frac{2k+1}{2U+1}}}}C^{UM_U}_{km_kl,-m_l}$  
    $\displaystyle \times (-1)^{l'-m'_l}{\textstyle{\sqrt{\frac{2k'+1}{2U'+1}}}}C^{U'M'_U}_{l'm'_lk',-m'_k} C^{T'M'_T}_{l'm'_ll,-m_l} C^{TM_T}_{km_kk',-m'_k}$  
  $\textstyle =$ $\displaystyle (-1)^{M'_T-M'_U}(-1)^{l}(-1)^{l'}{\textstyle{\sqrt{\frac{2k+1}{2U+1}}}}{\textstyle{\sqrt{\frac{2k'+1}{2U'+1}}}}(-1)^{l'+k'-U'}(-1)^{k+k'-T}$  
    $\displaystyle \times \sum_{m_lm'_lm_km'_k} C^{UM_U}_{km_kl,-m_l} C^{U'M'_U}_{k',-m'_kl'm'_l} C^{T'M'_T}_{l'm'_ll,-m_l} C^{TM_T}_{k',-m'_kkm_k}$  
  $\textstyle =$ $\displaystyle (-1)^{M'_T-M'_U}(-1)^{l+k-U'-T}{\textstyle{\sqrt{\frac{2k+1}{2U+1}}}}{\textstyle{\sqrt{\frac{2k'+1}{2U'+1}}}}$  
    $\displaystyle \times \sum_{m_lm'_lm_km'_k} C^{UM_U}_{km_klm_l} C^{U'M'_U}_{k'm'_kl'm'_l} C^{TM_T}_{k'm'_kkm_k} C^{T'M'_T}_{l'm'_llm_l}$  
  $\textstyle =$ $\displaystyle (-1)^{M'_T-M'_U}(-1)^{l+k-U'-T}\sqrt{(2k+1)(2k'+1)(2T'+1)(2T+1)}$  
    $\displaystyle \times \sum_{W'M'_W} C^{W'M'_W}_{TM_TT'M'_T} C^{W'M'_W}_{U'M'_UUM... ...rray}{rrr} l & k & U \\ l' & k' & U' \\ T' & T & W' \end{array}\right\} ,$ (125)

see Eq. 8.7(20) in Ref. [6].

We can now perform the summation over $M'_U$ and $M_U$, which gives the factor $(-1)^{U+U'-W}\delta_{WW'}\delta_{M_WM'_W}$ and allows for a summation over $W'$ and $M'_W$. After inserting all these results into Eq. (97), we obtain

$\displaystyle \tilde{\rho}^{uUu'U'W,J''M''}_{vJ'M'}(\vec{r}) \!\!\!\!$ $\textstyle =$ $\displaystyle \sum_{M_W\mu} C^{J'M'}_{WM_Wv\mu} (-1)^{U'} \sum_{{Nlj}\atop{N'l... ...yle{\frac{1}{2}}}\vert\vert\sigma_{v}\vert\vert{\textstyle{\frac{1}{2}}}\rangle$  
  $\textstyle \times$ $\displaystyle \sum_{k}{\textstyle{\frac{1}{\sqrt{2k+1}}}} F^{k}_{uUNl}(br) {\t... ...}} \langle\phi_{Nl j}\vert\vert\tilde{\rho}^{J''}\vert\vert\phi_{N'l'j'}\rangle$  
  $\textstyle \times$ $\displaystyle \sum_{k'}{\textstyle{\frac{1}{\sqrt{2k'+1}}}} F^{k'*}_{u'U'N'l'}(br)$  
  $\textstyle \times$ $\displaystyle (-1)^{j'+j-l+1-v} \sqrt{2}\sqrt{2j+1}\sqrt{2j+1}\sqrt{2j'+1}$  
    $\displaystyle \times \sum_{T'M'_T} C^{T'M'_T}_{v,-\mu J'',-M''} \left\{\begin{a... ...c{1}{2}}} & {\textstyle{\frac{1}{2}}} & v \\ l & l' & T' \end{array}\right\}$  
  $\textstyle \times$ $\displaystyle \sum_{TM_T} {\textstyle{\sqrt{\frac{(2k+1)(2k'+1)}{4\pi(2T+1)}}}} \times C^{T0}_{k0k'0} Y_{TM_T}(\theta,\phi)$  
  $\textstyle \times$ $\displaystyle (-1)^{M'_T+l+k-U'-T}\sqrt{(2k+1)(2k'+1)(2T'+1)(2T+1)}$  
    $\displaystyle \times C^{WM_W}_{TM_TT'M'_T} \left\{\begin{array}{rrr} l & k & U \\ l' & k' & U' \\ T' & T & W \end{array}\right\} (-1)^{U+U'-W}.$ (126)

Now we have to sum up products of three Clebsh-Gordan coefficients:

    $\displaystyle \sum_{M_WM'_T\mu} (-1)^{M'_T} C^{J'M'}_{WM_Wv\mu} C^{T'M'_T}_{v,-\mu J'',-M''} C^{WM_W}_{TM_TT'M'_T}$  
  $\textstyle =$ $\displaystyle \sum_{M_WM'_T\mu} (-1)^{M'_T} C^{J'M'}_{WM_Wv\mu} (-1)^{v+\mu}{\textstyle{\sqrt{\frac{2T'+1}{2J''+1}}}}C^{J''M''}_{v,-\mu T',-M'_T}$  
    $\displaystyle \times (-1)^{T'+M'_T}{\textstyle{\sqrt{\frac{2W+1}{2T+1}}}}C^{TM_T}_{WM_WT',-M'_T} (-1)^{W+T'-T}$  
  $\textstyle =$ $\displaystyle {\textstyle{\sqrt{\frac{2T'+1}{2J''+1}}}}(-1)^{T'}{\textstyle{\sqrt{\frac{2W+1}{2T+1}}}} (-1)^{W+T'-T}$  
    $\displaystyle \times \sum_{M_WM'_T\mu} (-1)^{v-\mu} C^{J''M''}_{v\mu T'M'_T} C^{TM_T}_{WM_WT'M'_T} C^{J'M'}_{WM_Wv,-\mu}$  
  $\textstyle =$ $\displaystyle {\textstyle{\sqrt{\frac{2T'+1}{2J''+1}}}}(-1)^{T'}{\textstyle{\sqrt{\frac{2W+1}{2T+1}}}} (-1)^{W+T'-T}$  
    $\displaystyle \times (-1)^{T'+W+J''+J'}\sqrt{2J''+1}\sqrt{2J'+1} C^{TM_T}_{J''M... ...} \left\{\begin{array}{rrr} v & T' & J'' \\ T & J' & W \end{array}\right\} ,$ (127)

see Eq. 8.7(15) in Ref. [6]. This gives:
$\displaystyle \tilde{\rho}^{uUu'U'W,J''M''}_{vJ'M'}(\vec{r})$ $\textstyle =$ $\displaystyle (-1)^{U'} \sum_{{Nlj}\atop{N'l'j'}} b^{3+u+u'} e^{-(br)^2} {\text... ...yle{\frac{1}{2}}}\vert\vert\sigma_{v}\vert\vert{\textstyle{\frac{1}{2}}}\rangle$  
  $\textstyle \times$ $\displaystyle \sum_{k}{\textstyle{\frac{1}{\sqrt{2k+1}}}} F^{k}_{uUNl}(br) {\t... ...}} \langle\phi_{Nl j}\vert\vert\tilde{\rho}^{J''}\vert\vert\phi_{N'l'j'}\rangle$  
  $\textstyle \times$ $\displaystyle \sum_{k'}{\textstyle{\frac{1}{\sqrt{2k'+1}}}} F^{k'*}_{u'U'N'l'}(br)$  
  $\textstyle \times$ $\displaystyle (-1)^{j'+j-l+1-v} \sqrt{2}\sqrt{2j+1}\sqrt{2j+1}\sqrt{2j'+1}$  
    $\displaystyle \times \sum_{T'} \left\{\begin{array}{rrr} j & j' & J'' \\ {\t... ...c{1}{2}}} & {\textstyle{\frac{1}{2}}} & v \\ l & l' & T' \end{array}\right\}$  
  $\textstyle \times$ $\displaystyle \sum_{TM_T} {\textstyle{\sqrt{\frac{(2k+1)(2k'+1)}{4\pi(2T+1)}}}} \times C^{T0}_{k0k'0} Y_{TM_T}(\theta,\phi)$  
  $\textstyle \times$ $\displaystyle (-1)^{l+k-U'-T}\sqrt{(2k+1)(2k'+1)(2T'+1)(2T+1)}$  
    $\displaystyle \times \left\{\begin{array}{rrr} l & k & U \\ l' & k' & U' \\ T' & T & W \end{array}\right\} (-1)^{U+U'-W}$  
  $\textstyle \times$ $\displaystyle {\textstyle{\sqrt{\frac{2T'+1}{2J''+1}}}}(-1)^{T'}{\textstyle{\sqrt{\frac{2W+1}{2T+1}}}} (-1)^{W+T'-T}(-1)^{T'+W+J''+J'}$  
    $\displaystyle \times \sqrt{2J''+1}\sqrt{2J'+1} C^{TM_T}_{J''M''J'M'} \left\{\begin{array}{rrr} v & T' & J'' \\ T & J' & W \end{array}\right\} .$ (128)

Finally we have:
$\displaystyle \tilde{\rho}^{uUu'U'W,J''M''}_{vJ'M'}(\vec{r})$ $\textstyle =$ $\displaystyle (-1)^{1+v} b^{3+u+u'} e^{-(br)^2} \langle{\textstyle{\frac{1}{2}}... ...}\vert\vert{\textstyle{\frac{1}{2}}}\rangle {\textstyle{\sqrt{\frac{1}{4\pi}}}}$  
  $\textstyle \times$ $\displaystyle \sum_{{Nljk}\atop{N'l'j'k'}} F^{k}_{uUNl}(br) \langle\phi_{Nl j}\vert\vert\tilde{\rho}^{J''}\vert\vert\phi_{N'l'j'}\rangle F^{k'}_{u'U'N'l'}(br)$  
  $\textstyle \times$ $\displaystyle \sum_{T'TM_T}(-1)^{j'+j}(-1)^{k}(-1)^{T+T'}(-1)^{U-W}C^{T0}_{k0k'0}$  
  $\textstyle \times$ $\displaystyle {\textstyle{\sqrt{\frac{(2T'+1)(2T'+1)(2k+1)(2k'+1)(2J'+1)(2W+1)(2j+1)(2j'+1)}{(2T+1)}}}}$  
  $\textstyle \times$ $\displaystyle \left\{\begin{array}{rrr} j & j' & J'' \\ {\textstyle{\frac{1}... ...t\} \left\{\begin{array}{rrr} v & T' & J'' \\ T & J' & W \end{array}\right\}$  
  $\textstyle \times$ $\displaystyle C^{TM_T}_{J'M'J''M''} Y_{TM_T}(\theta,\phi) ,$ (129)

where we have also used Eq. (90). This gives expression (98) and definitions of radial form factors (99) and coefficients (100).

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Next: Derivation of Eq. (113) Up: Solution of self-consistent equations Previous: Calculation of coefficients (72)
Jacek Dobaczewski 2010-01-30