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Matrix Elements Between Complex Configurations

According to the approach P2 [4], a general expression of a submatrix element for any two-particle operator between functions with u open shells, can be written as follows:


$\displaystyle {(\psi _u^{bra}\left( LS\right) \vert\vert G\vert\vert\psi _u^{ket}\left( L^{\prime }S^{\prime
}\right) )}$
  $\textstyle =$ $\displaystyle \displaystyle {\sum_{n_il_i,n_jl_j,n_i^{\prime }l_i^{\prime },n_j...
...{\prime } \right)
\vert\vert\psi _u^{ket}\left( L^{\prime }S^{\prime
}\right) )$  
  $\textstyle =$ $\displaystyle \displaystyle {\sum_{n_il_i,n_jl_j,n_i^{\prime }l_i^{\prime },n_j...
...rime }\lambda _i^{\prime
},n_j^{\prime }\lambda _j^{\prime },\Xi \right) \times$  
    $\displaystyle T\left(
n_i\lambda _i,n_j\lambda _j,n_i^{\prime }\lambda _i^{\pri...
..._i^{\prime },\lambda _j^{\prime
},\Lambda ^{bra},\Lambda ^{ket},\Gamma \right),$ (1)

where $\lambda \equiv l,s$, $\Lambda _l^{bra}\equiv
\left( L_i,L_j,L_i^{\prime },L_j^{\prime }\right) ^{bra}$, $\Lambda
_s^{bra}\equiv \left( S_i,S_j,S_i^{\prime },S_j^{\prime }\right) ^{bra}$ and $\Gamma$ refers to the array of coupling parameters connecting the recoupling matrix to the submatrix element.

So, to calculate the spin-angular part of a submatrix element of this type, one has to obtain:

  1. Recoupling matrix $R\left( \lambda _i,\lambda _j,\lambda _i^{\prime },\lambda _j^{\prime },
\Lambda ^{bra},\Lambda ^{ket},\Gamma \right) $.

  2. Submatrix elements $T\left( n_i\lambda _i,n_j\lambda _j,n_i^{\prime }\lambda _i^{\prime
},n_j^{\prime }\lambda _j^{\prime },\Lambda ^{bra},\Lambda ^{ket},\Xi
,\Gamma \right) $.

  3. Phase factor $\Delta $.

  4. $\Theta ^{\prime }\left(
n_i\lambda _i,n_j\lambda _j,n_i^{\prime }\lambda _i^{\prime },n_j^{\prime
}\lambda _j^{\prime },\Xi \right) $.

Some important points to note are the following:

1. The recoupling matrices $R\left( \lambda _i,\lambda _j,\lambda _i^{\prime },\lambda _j^{\prime },
\Lambda ^{bra},\Lambda ^{ket},\Gamma \right) $ in our approach are much simpler than in other known approaches. We have obtained their analytical expressions in terms of just 6j- and 9j-coefficients. That is why we choose a special form of operator in second quantization, where second quantization operators acting upon the same shell are tensorially coupled together.

2. The tensorial part of a two-particle operator is expressed in terms of (products of) operators of the type $A^{\left(
kk\right) }\left( n\lambda ,\Xi \right) $, $B^{\left( kk\right) }(n\lambda
,\Xi )$, $C^{\left( kk\right) }(n\lambda ,\Xi )$, $D^{\left( ls\right) }$, $%%
E^{\left( kk\right) }(n\lambda ,\Xi )$. Their explicit expressions are (2)-(6):


\begin{displaymath}
a_{m_q}^{\left( q\lambda \right) },
\end{displaymath} (2)


\begin{displaymath}
\left[ a_{m_{q1}}^{\left( q\lambda \right) }\times
a_{m_{q2}...
...\lambda \right) }\right] ^{\left( \kappa _1\sigma
_1\right) },
\end{displaymath} (3)


\begin{displaymath}
\left[ a_{m_{q1}}^{\left( q\lambda \right) }\times \left[
a_...
...sigma _1\right) }\right] ^{\left( \kappa
_2\sigma _2\right) },
\end{displaymath} (4)


\begin{displaymath}
\left[ \left[ a_{m_{q1}}^{\left( q\lambda \right) }\times
a_...
...\lambda \right) }\right] ^{\left(
\kappa _2\sigma _2\right) },
\end{displaymath} (5)


\begin{displaymath}
\left[ \left[ a_{m_{q1}}^{\left( q\lambda \right) }\times
a_...
...left( \kappa _2\sigma
_2\right) }\right] ^{\left( kk\right)} .
\end{displaymath} (6)

We denote their submatrix elements by $T\left( n_i\lambda _i,n_j\lambda _j,n_i^{\prime }\lambda _i^{\prime
},n_j^{\prime }\lambda _j^{\prime },\Lambda ^{bra},\Lambda ^{ket},\Xi
,\Gamma \right) $. The parameter $\Gamma$ represents the whole array of parameters connecting the recoupling matrix $R\left( \lambda _i,\lambda _j,\lambda _i^{\prime },\lambda _j^{\prime },
\Lambda ^{bra},\Lambda ^{ket},\Gamma \right) $ to the submatrix element $T\left( n_i\lambda _i,n_j\lambda _j,n_i^{\prime }\lambda _i^{\prime
},n_j^{\prime }\lambda _j^{\prime },\Lambda ^{bra},\Lambda ^{ket},\Xi
,\Gamma \right) $. It is worth noting that each of the tensorial quantities (2)-(6) act upon one and the same shell. So, all the advantages of tensor algebra and the quasispin formalism may be efficiently exploited in the process of their calculation.

We obtain the submatrix elements of operator (2) by using straightforwardly the Wigner-Eckart theorem in quasispin space:


\begin{displaymath}
\begin{array}[b]{c}
\left( l^N\;\alpha QLS\vert\vert a_{m_q}...
...{\prime
}Q^{\prime }L^{\prime }S^{\prime }\right) ,
\end{array}\end{displaymath} (7)

where the last multiplier in (7) is the so-called reduced coefficients of fractional parentage (RCFPs) and we use a shorthand notation $(2k+1)\cdot ...\equiv [k,...]$.

The value of a submatrix element of operator (3) is obtained by basing ourselves on (33), (34) in P1 [3]. In the other three cases (4), (5), (6) we obtain them by using (2.28) of Jucys and Savukynas [8]:


\begin{displaymath}
\begin{array}[b]{c}
(nl^N\;\alpha QLS\vert\vert\left[ F^{\le...
...me } & S & S^{\prime \prime }
\end{array}\right\} ,
\end{array}\end{displaymath} (8)

where $F^{\left( \kappa _1\sigma _1\right) }\left( n\lambda \right) $, $%%
G^{(\kappa _2\sigma _2)}\left( n\lambda \right) $ is one of (2) or (3) and the submatrix elements correspondingly are defined by (7) and (33), (34) in P1 [3]. $N^{\prime \prime }$ is defined by second quantization operators occurring in $F^{\left( \kappa _1\sigma _1\right) }\left( n\lambda \right) $ and $G^{(\kappa _2\sigma _2)}\left(
n\lambda \right) $.

As is seen, by using this approach, the calculation of the angular parts of matrix elements between functions with u open shells ends up as a calculation of submatrix elements of tensors (2), (3) within single shell of equivalent electrons. As these completely reduced (reduced in the quasispin, orbital and spin spaces) submatrix elements do not depend on the occupation number of the shell, the tables for them are reduced considerably in comparison with the tables of analogous submatrix elements of tensorial quantities $U^k,$ $V^{k_1k_2}$ (Jucys and Savukynas [8] or Cowan [9]) and the tables of fractional parentage coefficients (CFP). That is why the expressions obtained are very useful in practical calculations. This is extremely important for the f - subshell, where the number of CFPs for $f^1$ - $f^{14}$ equals 54408, whereas the number of RCFP, taking into account the transposition symmetry property of RCFP is only 14161 - of which only 3624 are nonzero.

We do not present details on obtaining phase factors $\Delta $ and $\Theta ^{\prime }\left(
n_i\lambda _i,n_j\lambda _j,n_i^{\prime }\lambda _i^{\prime },n_j^{\prime
}\lambda _j^{\prime },\Xi \right) $, since no essential generalizations may be made here; those are possible only after a particular operator is chosen (for more details see P2 [4], P3 [5]).


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Next: The Electrostatic Electron Interaction, Up: Notations and Methodology of Previous: Notations and Methodology of
2001-12-07