Matrix elements of the Hamiltonian (7) in the spherical HO basis

For the Hamiltonian in the form given by Eq. (7), a similar derivation gives the matrix elements in the spherical HO basis that can be written as:

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

$$\times \int r^2{\rm d}\,r b^{3+n'} e^{-2(br)^{2}} F_{00N'l'}^{l'}(br) \sum_{n'L'k}F_{n'L'Nl}^{k}(br)$$

$$\times \sum_{v'J'T}{U}_{n'L'v'J'}^{TJ''}(br) \langle{\textstyle{\frac{1}... ...igma_{v'}\vert\vert{\textstyle{\frac{1}{2}}}\rangle h_{l'j',v'T}^{ljkL',J'J''}$$ (108)

with the reduced form factors ${U}_{n'L'v'J'}^{TJ''}(br)$ given in Eq. (107), and

$$h_{l'j',v'T}^{ljkL',J'J''} = \frac{1}{2\sqrt{\pi}}\left(-1\right)^{J''+l+v'+l'+L'} \frac{\sqrt{\left(2k+1\right)\left(2j'+1\right)\left(2j+1\right)}}{\sqrt{\left(2l'+1\right)}}$$

$$\times \left(-1\right)^{T}\left(2T+1\right)C_{k0T0}^{l'0}$$

$$\times \sum_{T'}\left(-1\right)^{T'}\left(2T'+1\right)\left\{ \begin{arr... ...ac{1}{2} & \frac{1}{2} & v'\\ j' & j & J''\\ l' & l & T'\end{array}\right\}$$

$$\times \left\{ \begin{array}{ccc} T' & l & l'\\ k & T & L'\end{array}\... ...t\} \left\{ \begin{array}{ccc} J'' & T & J'\\ L' & v'& T'\end{array}\right\}.$$ (109)