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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:

    $\displaystyle \langle\phi_{N'l'j'}\vert\vert\tilde{h}^{I''}(\tilde{\rho}^{J''M''})\vert\vert\phi_{Nlj}\rangle =
\delta_{I''J''}$  
  $\textstyle \times$ $\displaystyle \int r^2{\rm d}\,r b^{3+n'} e^{-2(br)^{2}} F_{00N'l'}^{l'}(br)
\sum_{n'L'k}F_{n'L'Nl}^{k}(br)$  
  $\textstyle \times$ $\displaystyle \sum_{v'J'T}{U}_{n'L'v'J'}^{TJ''}(br)
\langle{\textstyle{\frac{1}...
...igma_{v'}\vert\vert{\textstyle{\frac{1}{2}}}\rangle
h_{l'j',v'T}^{ljkL',J'J''}$ (108)

with the reduced form factors ${U}_{n'L'v'J'}^{TJ''}(br)$ given in Eq. (107), and
$\displaystyle h_{l'j',v'T}^{ljkL',J'J''}$ $\textstyle =$ $\displaystyle \frac{1}{2\sqrt{\pi}}\left(-1\right)^{J''+l+v'+l'+L'}
\frac{\sqrt{\left(2k+1\right)\left(2j'+1\right)\left(2j+1\right)}}{\sqrt{\left(2l'+1\right)}}$  
  $\textstyle \times$ $\displaystyle \left(-1\right)^{T}\left(2T+1\right)C_{k0T0}^{l'0}$  
  $\textstyle \times$ $\displaystyle \sum_{T'}\left(-1\right)^{T'}\left(2T'+1\right)\left\{ \begin{arr...
...ac{1}{2} & \frac{1}{2} & v'\\
j' & j & J''\\
l' & l & T'\end{array}\right\}$  
  $\textstyle \times$ $\displaystyle \left\{ \begin{array}{ccc}
T' & l & l'\\
k & T & L'\end{array}\...
...t\} \left\{ \begin{array}{ccc}
J'' & T & J'\\
L' & v'& T'\end{array}\right\}.$ (109)


next up previous
Next: The total potential energy Up: The NLO potentials, fields, Previous: Matrix elements of the
Jacek Dobaczewski 2010-01-30