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SAI_NORE

This library is for calculating the angular parts of matrix elements for a scalar two-particle operator. It contains 18 subroutines. Most of the subroutines from this module use common blocks CONSTS and MEDEFN from MCHF atomic structure package [1,11]. The calculations are performed according to the methodology P2 [4].

COULOMBLS Investigates the two-electron submatrix elements of electrostatic interaction

\begin{displaymath}\left( n_i\lambda _in_j\lambda _j\vert\vert g_{Coulomb}^{\lef...
... }\lambda _i^{\prime }n_j^{\prime }\lambda _j^{\prime }\right),\end{displaymath}

according to the formula (9) of P1 [3]. The values of these matrix elements are needed because of (see P2 [4]):


\begin{displaymath}
\Theta \left( \Xi \right) \sim \left( n_i\lambda _in_j\lambd...
...a _{i^{\prime }}n_{j^{\prime
}}\lambda _{j^{\prime }}\right) .
\end{displaymath} (41)

The value of the output parameter AA of this subroutine is:

\begin{displaymath}AA=2\left[ k\right] ^{1/2}\left( l_i\vert\vert C^{\left( k\ri...
...\vert\vert C^{\left( k\right) }\vert\vert l_j^{\prime }\right).\end{displaymath}

Table 3 lists the expressions used by each of the subroutines NONRELAT1, NONRELAT2, NONRELAT31, NONRELAT32, NONRELAT33, NONRELAT41, NONRELAT51, NONRELAT52, NONRELAT53. The numbering of expressions is the same as in paper P2 [4], where all these expressions are presented. As the structure of all the subroutines mentioned earlier is the same, and only different expressions are used and different subroutines are called, we will discuss in more detail only one of these subroutines.


Table 3: Scheme of the expressions for matrix elements of two-particle scalar operator (like Coulomb interaction).
Dis. $\widehat{G}$ $\widehat{G}(T)$ $\alpha$ $\beta$ $\gamma$ $\delta$ $ \tilde \Theta$ $R$ $\Delta $
NONRELAT1              
$\alpha\alpha\alpha\alpha$ (47) (5) (38),(35) - - - (48),(49) (18) (41)
$\alpha\beta\alpha\beta$ (50) (6) (35) (35) - - (51) (22) (41)
$\beta\alpha\beta\alpha$ (50) (6) (35) (35) - - (51) (22) (41)
$\alpha\beta\beta\alpha$ (54) (6) (35) (35) - - (55) (22) (41)
$\beta\alpha\alpha\beta$ (54) (6) (35) (35) - - (55) (22) (41)
NONRELAT2              
$\alpha\alpha\beta\beta$ (52) (6) (35) (35) - - (53) (22) (41)
NONRELAT31              
$\beta\alpha\alpha\alpha$ (56) (6) (36) (34) - - (58) (22) (42)
$\alpha\beta\alpha\alpha$ (56) (6) (36) (34) - - (59) (22) (42)
NONRELAT32              
$\beta\beta\beta\alpha$ (60) (6) (34) (37) - - (62) (22) (42)
$\beta\beta\alpha\beta$ (60) (6) (34) (37) - - (63) (22) (42)
NONRELAT33              
$\beta\gamma\alpha\gamma$ (50) (7) (34) (34) (35) - (51) (26) (42)
$\gamma\beta\gamma\alpha$ (50) (7) (34) (34) (35) - (51) (26) (42)
$\gamma\beta\alpha\gamma$ (54) (7) (34) (34) (35) - (55) (26) (42)
$\beta\gamma\gamma\alpha$ (54) (7) (34) (34) (35) - (55) (26) (42)
NONRELAT41              
$\gamma\gamma\alpha\beta$ (52) (7) (34) (34) (35) - (53) (26) (42)
$\gamma\gamma\beta\alpha$ (52) (7) (34) (34) (35) - (53) (26) (42)
$\alpha\beta\gamma\gamma$ (52) (7) (34) (34) (35) - (53) (26) (42)
$\beta\alpha\gamma\gamma$ (52) (7) (34) (34) (35) - (53) (26) (42)
NONRELAT51              
$\alpha\beta\gamma\delta$ (52) (8) (34) (34) (34) (34) (53) (33) (43)
$\beta\alpha\gamma\delta$ (52) (8) (34) (34) (34) (34) (53) (33) (43)
$\alpha\beta\delta\gamma$ (52) (8) (34) (34) (34) (34) (53) (33) (43)
$\beta\alpha\delta\gamma$ (52) (8) (34) (34) (34) (34) (53) (33) (43)
$\gamma\delta\alpha\beta$ (52) (8) (34) (34) (34) (34) (53) (33) (43)
$\gamma\delta\beta\alpha$ (52) (8) (34) (34) (34) (34) (53) (33) (43)
$\delta\gamma\alpha\beta$ (52) (8) (34) (34) (34) (34) (53) (33) (43)
$\delta\gamma\beta\alpha$ (52) (8) (34) (34) (34) (34) (53) (33) (43)
NONRELAT52              
$\alpha\gamma\beta\delta$ (50) (8) (34) (34) (34) (34) (51) (33) (43)
$\alpha\gamma\delta\beta$ (54) (8) (34) (34) (34) (34) (55) (33) (43)
$\gamma\alpha\delta\beta$ (50) (8) (34) (34) (34) (34) (51) (33) (43)
$\gamma\alpha\beta\delta$ (54) (8) (34) (34) (34) (34) (55) (33) (43)
$\beta\delta\alpha\gamma$ (50) (8) (34) (34) (34) (34) (51) (33) (43)
$\delta\beta\gamma\alpha$ (50) (8) (34) (34) (34) (34) (51) (33) (43)
$\beta\delta\gamma\alpha$ (54) (8) (34) (34) (34) (34) (55) (33) (43)
$\delta\beta\alpha\gamma$ (54) (8) (34) (34) (34) (34) (55) (33) (43)
NONRELAT53              
$\alpha\delta\beta\gamma$ (50) (8) (34) (34) (34) (34) (51) (33) (43)
$\delta\alpha\gamma\beta$ (50) (8) (34) (34) (34) (34) (51) (33) (43)
$\alpha\delta\gamma\beta$ (54) (8) (34) (34) (34) (34) (55) (33) (43)
$\delta\alpha\beta\gamma$ (54) (8) (34) (34) (34) (34) (55) (33) (43)
$\beta\gamma\alpha\delta$ (50) (8) (34) (34) (34) (34) (51) (33) (43)
$\gamma\beta\delta\alpha$ (50) (8) (34) (34) (34) (34) (51) (33) (43)
$\beta\gamma\delta\alpha$ (54) (8) (34) (34) (34) (34) (55) (33) (43)
$\gamma\beta\alpha\delta$ (54) (8) (34) (34) (34) (34) (55) (33) (43)

NONRELAT1 is meant for finding angles for the distributions $\alpha\alpha\alpha\alpha$, $\alpha\beta\alpha\beta$ and $\beta\alpha\beta\alpha$.

In the $\alpha\alpha\alpha\alpha$ case, the program uses expression (5) from P2 [4]. In this case


\begin{displaymath}
\widehat{G}\left( I\right) \sim \displaystyle
{\sum_{\kappa ...
...p,-p}^{\left( kk\right) }\left( n\lambda ,\Xi \right) \right],
\end{displaymath} (42)

where


\begin{displaymath}
A_{a}^{\left( 00\right) }\left( n\lambda ,\Xi \right)=
\left...
...{\left( \kappa _2\sigma _2\right) }\right] ^{\left(00\right)},
\end{displaymath} (43)


\begin{displaymath}
A_{(b)p,-p}^{\left( kk\right) }\left( n\lambda ,\Xi \right)=...
...left( l_\alpha
s\right) }\right] _{p,-p}^{\left( kk \right) },
\end{displaymath} (44)

and


\begin{displaymath}
\begin{array}[b]{c}
\Theta_{IIa} \left( n\lambda ,\Xi \right...
...ha \lambda _\alpha
n_\alpha \lambda _\alpha \right)
\end{array}\end{displaymath} (45)

and


\begin{displaymath}
\begin{array}[b]{c}
\Theta_{IIb} \left( n\lambda ,\Xi \right...
...& \sigma _2 & k \\
s & s & s
\end{array}\right\} .
\end{array}\end{displaymath} (46)

The value of the reduced matrix element of operator (43) is found by subroutine WWLS1, and that of (44) by W1 (see section SAI_SQLS1). The value of coefficient $\Theta_{IIa} \left( n\lambda ,\Xi \right)$ is calculated by COULOMBLS subroutine, because this coefficient, to the accuracy of a factor and a phase, is equal to that part of the two-electron submatrix elements of electrostatic interaction, which this subroutine is calculating. The coefficient $\Theta_{IIb} \left( n\lambda ,\Xi \right)$ is found by COULOMBLS and SIXJ, and the recoupling matrix is investigated by RECOUP0 (see section SAI_RECLS).

For the distributions $\alpha\beta\alpha\beta$ and $\beta\alpha\beta\alpha$ the subroutine NONRELAT1 uses (6) of P2 [4], keeping in mind that $\Theta \left( n_\alpha \lambda _\alpha ,n_\beta \lambda _\beta ,\Xi \right)$ is expressed as (51) of P2 [4] and tensorial parts $B^{\left( \kappa _{12}\sigma _{12}\right) }\left( n_\alpha \lambda _\alpha
,\Xi \right)$, $C^{\left( \kappa _{12}^{\prime }\sigma _{12}^{\prime
}\right) }\left( n_\beta \lambda _\beta ,\Xi \right)$ are equal to (35) from P2 [4]. The coefficients $\Theta \left( n_\alpha \lambda _\alpha ,n_\beta \lambda _\beta ,\Xi \right)$ are investigated by COULOMBLS, the coefficients $B^{\left( \kappa _{12}\sigma _{12}\right) }\left( n_\alpha \lambda _\alpha
,\Xi \right)$ and $C^{\left( \kappa _{12}^{\prime }\sigma _{12}^{\prime
}\right) }\left( n_\beta \lambda _\beta ,\Xi \right)$ are found by W1W2LS from the SAI_SQLS1 library, and the recoupling matrix is calculated by RECOUP2.

For the distributions $\alpha\beta\beta\alpha$ and $\beta\alpha\alpha\beta$ the subroutine NONRELAT1 uses (6) of P2 [4], keeping in mind that $\Theta \left( n_\alpha \lambda _\alpha ,n_\beta \lambda _\beta ,\Xi \right)$ is expressed as (55) of P2 [4] and tensorial parts $B^{\left( \kappa _{12}\sigma _{12}\right) }\left( n_\alpha \lambda _\alpha
,\Xi \right)$, $C^{\left( \kappa _{12}^{\prime }\sigma _{12}^{\prime
}\right) }\left( n_\beta \lambda _\beta ,\Xi \right)$ are equal to (35) from P2 [4]. The coefficients $\Theta \left( n_\alpha \lambda _\alpha ,n_\beta \lambda _\beta ,\Xi \right)$ are investigated by COULOMBLS and SIXJ, the coefficients $B^{\left( \kappa _{12}\sigma _{12}\right) }\left( n_\alpha \lambda _\alpha
,\Xi \right)$ and $C^{\left( \kappa _{12}^{\prime }\sigma _{12}^{\prime
}\right) }\left( n_\beta \lambda _\beta ,\Xi \right)$ are found by W1W2LS from the SAI_SQLS1 library, and the recoupling matrix is calculated by RECOUP2.


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Next: SAI_DUDU Up: Description of the Libraries Previous: The subroutine RUMT67
2001-12-07