Matrix elements of the Hamiltonian (60) in the spherical HO basis

In the spherical HO basis of Eq. (79), we can calculate the matrix elements of the single-particle Hamiltonian (60) in the following way:

$$\langle N'l'j'm'_j\vert\tilde{h}(\tilde{\rho}^{J''M''})\vert Nljm... ...IM_I'} \sum_{M'_L\mu'} \int r^2{\rm d}\,r\int {\rm d}\Omega\sum_{\sigma'\sigma}$$

$$\sum_{M'}C^{J'M'}_{L'M'_Lv'\mu'} {\textstyle{\frac{(-1)^{J'-M'}}{... ...'} C^{L'M'_L}_{IM_II'M'_I} \langle\sigma'\vert\sigma_{v'\mu'}\vert\sigma\rangle$$

$$\phi^*_{N'l'j'm'_j}(r,\theta,\phi,\sigma')\, D_{m'I'M'_I}\tilde{U... ...'M''}_{n'L'v'J',-M'}(\vec{r})D_{mIM_I}\, \phi_{Nljm_j}(r,\theta,\phi,\sigma) ,$$ (103)

where $\tilde{h}(\tilde{\rho}^{J''M''})$ denotes the field calculated for the one-multipole density matrix $\tilde{\rho}^{J''M''}$ (91). The integration by parts now gives

$$\langle N'l'j'm'_j\vert\tilde{h}(\tilde{\rho}^{J''M''})\vert Nljm... ...IM_I'} \sum_{M'_L\mu'} \int r^2{\rm d}\,r\int {\rm d}\Omega\sum_{\sigma'\sigma}$$

$$\sum_{M'}C^{J'M'}_{L'M'_Lv'\mu'} {\textstyle{\frac{(-1)^{J'-M'}}{... ...II'M'_I} \langle\sigma'\vert\sigma_{v'\mu'}\vert\sigma\rangle (-1)^{m'+I'-M'_I}$$

$$\tilde{U}^{J''M''}_{n'L'v'J',-M'}(\vec{r})\, \left(D_{m'I',-M'_I}... ...\sigma')\right)^* \left(D_{mIM_I} \phi_{Nljm_j}(r,\theta,\phi,\sigma)\right) ,$$

where we have used the hermitian-conjugation property (32) of the differential operators.

By inserting potentials (104) and derivatives of spherical wavefunctions (83) we have:

$$\langle N'l'j'm'_j\vert\tilde{h}(\tilde{\rho}^{J''M''})\vert Nljm... ...IM_I'} \sum_{M'_L\mu'} \int r^2{\rm d}\,r\int {\rm d}\Omega\sum_{\sigma'\sigma}$$

$$\sum_{M'}C^{J'M'}_{L'M'_Lv'\mu'} {\textstyle{\frac{(-1)^{J'-M'}}{... ...II'M'_I} \langle\sigma'\vert\sigma_{v'\mu'}\vert\sigma\rangle (-1)^{m'+I'-M'_I}$$

$$\times \sum_{TM_T} \tilde{U}_{n'L'v'J'}^{TJ''}(br) e^{-(br)^2} C^{TM_T}_{J',-M'J''M''} Y_{TM_T}(\theta,\phi)$$

$$\times b^{m'+3/2}\sum_{k'm'_k}\sum_{m'_lm'_s} {\textstyle{\frac{1... ...}} C^{k'm'_k}_{l'm'_lI',-M'_I}C^{j'm'_j}_{l'm'_l{\textstyle{\frac{1}{2}}}m'_s}$$

$$\times F^{k'*}_{m'I'N'l'}(br)e^{-{\textstyle{\frac{1}{2}}}(br)^2}Y^*_{k'm'_k}(\theta,\phi)\chi_{{\textstyle{\frac{1}{2}}}m'_s}(\sigma')$$

$$\times b^{m+3/2}\sum_{km_k}\sum_{m_lm_s} {\textstyle{\frac{1}{\sqrt{2k+1}}}} C^{km_k}_{lm_lIM_I}C^{jm_j}_{lm_l{\textstyle{\frac{1}{2}}}m_s}$$

$$\times F^{k}_{mINl}(br)e^{-{\textstyle{\frac{1}{2}}}(br)^2}Y_{km_k}(\theta,\phi)\chi_{{\textstyle{\frac{1}{2}}}m_s}(\sigma) ,$$ (104)

The angular part can be integrated explicitly, by using the multiplication law of spherical harmonics (Eq. 5.6(9) in Ref. [6]), and summations over $\sigma$ and $\sigma'$ can be performed as in Eqs. (94)-(96). This gives

$$\langle N'l'j'm'_j\vert\tilde{h}(\tilde{\rho}^{J''M''})\vert Nljm... ...= \sum_{n'L'v'J'}\sum_{mIm'I'}\sum_{M_IM_I'} \sum_{M'_L\mu'} \int r^2{\rm d}\,r$$

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

$$\times \sum_{TM_T} \sum_{k'm'_k}\sum_{m'_lm'_s} \sum_{km_k}\sum_{m_lm_s}... ...tstyle{\frac{(2T+1)(2k+1)}{4\pi(2k'+1)}}}} C^{k'0}_{T0k0} C^{k'm_k'}_{TM_Tkm_k}$$

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

$$\times C^{TM_T}_{J',-M'J''M''} {\textstyle{\frac{1}{\sqrt{2k'+1}}}} C^{... ...sqrt{2k+1}}}} C^{km_k}_{lm_lIM_I}C^{jm_j}_{lm_l{\textstyle{\frac{1}{2}}}m_s} ,$$ (105)

where we have used condition (56).

We may now proceed by calculating the reduced matrix element of the field:

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

$$\sum_{m_jm'_jM''_I}\frac{1}{\sqrt{2j'+1}} C^{j'm'_j}_{jm_jI''M''_I} \langle N'l'j'm'_j\vert\tilde{h}(\tilde{\rho}^{J''M''})\vert Nljm_j\rangle .$$ (106)

Again, after a lengthy but straightforward derivation presented in C, we obtain the following result:

$$\langle\phi_{N'l'j'}\vert\vert\tilde{h}^{I''}(\tilde{\rho}^{J''M''})\vert\vert\phi_{Nlj}\rangle = \delta_{I''J''} (-1)^{J''}$$

$$\times \sum_{n'L'v'J'}\sum_{mIm'I'} K^{n'L'}_{mIm'I'} (-1)^{v'}\langle{\... ...c{1}{2}}}\rangle{\textstyle{\sqrt{\frac{1}{4\pi}}}} \sum_{Tkk'} (-1)^{T} (2T+1)$$

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

$$\times B^{ljkI,L'J''}_{l'j'k'I',v'TJ'} ,$$ (107)

where we see the same numerical coefficients (100) that already appeared in Eq. (99).