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Case 3. Calculating the matrix elements of $V^{k}$ operator

In this section it is demonstrated how to calculate the reduced $V^{11}$ operator, using the subroutine RWLS (see in Section 3.2), which calculates the reduced matrix elements (35). For that purpose the relation (16.34) from Rudzikas [14] is used. While using the $U^{k}$ and $V^{k1}$ tables, one must pay attention to various phase conventions used in the literature. In addition, small differences in the definitions of $U^{k}$ occur. Some authors, Karazija et al [20] among them, tabulate the submatrix elements

\begin{displaymath}
(l^{N} \alpha SL \vert\vert U^{k} \vert\vert l^{N} \alpha ^{\prime}S^{\prime}L^{\prime}),
\end{displaymath} (47)

others, like Nielson and Koster [18] or Cowan [9], tabulate
\begin{displaymath}
(l^{N} \alpha L \vert\vert U^{k} \vert\vert l^{N} \alpha ^{\prime}L^{\prime}),
\end{displaymath} (48)

although they use the notation of (47). Meanwhile, the relation between these two coefficients is:
\begin{displaymath}
(l^{N} \alpha SL \vert\vert U^{k} \vert\vert l^{N} \alpha ^{...
...\vert\vert U^{k} \vert\vert l^{N} \alpha ^{\prime}L^{\prime}).
\end{displaymath} (49)

The submatrix elements are defined as (47) if we use the relations between matrix elements of $W^{(k_1k_2k_3)}$ and $U^{k}$ as presented in the Rudzikas monograph [14]. Reduced matrix elements of the operator $V^{11}$ $(f^{7} ^{6}P^{0} \vert\vert V^{11} \vert\vert f^{7} ^{4}S^{1})$ are calculated in the example. The numerical value of this reduced matrix element is taken from the Nielson and Koster [18] tables

\begin{displaymath}(f^{7} ^{6}P^{0} \vert\vert V^{11} \vert\vert f^{7} ^{4}S^{1}) = -\sqrt{\frac{2}{7}}\end{displaymath}

and agree with our value.


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Next: Table 4. EXAMPLES RUN Up: Examples Previous: Case 2. Calculating the
2001-12-07