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The Electrostatic Electron Interaction, Spin-Spin and Spin-Other-Orbit Operators

The electrostatic (Coulomb) electron interaction operator $H^{Coulomb}$ itself contains the tensorial structure


\begin{displaymath}
\begin{array}[b]{c}
H^{Coulomb} \equiv
\displaystyle {\sum_{k}} H_{Coulomb}^{(kk0,000)}
\end{array}\end{displaymath} (9)

and its submatrix element is:


\begin{displaymath}
\begin{array}[b]{c}
\left( n_i\lambda _in_j\lambda _j\left\V...
...me }},n_jl_j n_{j^{\prime
}}l_{j^{\prime }}\right).
\end{array}\end{displaymath} (10)

The spin-spin operator $H^{ss}$ itself contains tensorial structure of two different types, summed over k:


\begin{displaymath}
\begin{array}[b]{c}
H^{ss} \equiv
\displaystyle {\sum_{k}}
\...
...k+1 k-1 2,112)} + H_{ss}^{(k-1 k+1 2,112)} \right].
\end{array}\end{displaymath} (11)

Their submatrix elements are:


\begin{displaymath}
\begin{array}[b]{c}
\left( n_i\lambda _in_j\lambda _j\left\V...
...{i^{\prime }}n_{j^{\prime
}}l_{j^{\prime }}\right),
\end{array}\end{displaymath} (12)


\begin{displaymath}
\begin{array}[b]{c}
\left( n_i\lambda _in_j\lambda _j\left\V...
...{j^{\prime }}n_{i^{\prime
}}l_{i^{\prime }}\right),
\end{array}\end{displaymath} (13)

where we use a shorthand notation $\left( 2k+3\right) ^{\left( 5\right) } \equiv \left( 2k+3\right)
\left( 2k+2\right)\left( 2k+1\right)\left( 2k\right)\left( 2k-1\right)$ and radial integral (12), (13) is defined as in Glass and Hibbert [16]:


\begin{displaymath}
\begin{array}[b]{c}
N^k\left( n_il_in_jl_j,n_{i^{\prime }}l_...
...right) P_{j^{\prime }}\left( r_2\right) dr_1dr_{2},
\end{array}\end{displaymath} (14)

where $\epsilon (x)$ is a Heaviside step-function,


\begin{displaymath}
\epsilon (x)=\left\{
\begin{array}{ll}
1 ; & \mbox{ for } x>0,  0 ; & \mbox{ for } x\leq 0.
\end{array}\right.
\end{displaymath} (15)

The spin-other-orbit operator $H^{soo}$ itself contains tensorial structure of six different types, summed over $k$:

\begin{displaymath}
\begin{array}[b]{c}
H^{sso} \equiv
\displaystyle {\sum_{k}}
...
...^{(k+1 k 1,101)} + H_{sso}^{(k+1 k 1,011)}
\right].
\end{array}\end{displaymath} (16)

Their submatrix elements are:


\begin{displaymath}
\begin{array}[b]{c}
\left( n_i\lambda _in_j\lambda _j\left\V...
...{j^{\prime }}n_{i^{\prime }}l_{i^{\prime }}\right),
\end{array}\end{displaymath} (17)


\begin{displaymath}
\begin{array}[b]{c}
\left( n_i\lambda _in_j\lambda _j\left\V...
...e }}n_{j^{\prime }}l_{j^{\prime
}}\right) \right\},
\end{array}\end{displaymath} (18)


\begin{displaymath}
\begin{array}[b]{c}
\left( n_i\lambda _in_j\lambda _j\left\V...
...i^{\prime }}n_{j^{\prime }}l_{j^{\prime }}\right) .
\end{array}\end{displaymath} (19)

The radial integrals in (17) - (19) are (see Glass and Hibbert [16]):


\begin{displaymath}
\begin{array}[b]{c}
V^k\left( n_il_in_jl_j,n_{i^{\prime }}l_...
...1\right) P_{j^{\prime }}\left( r_2\right)
dr_1dr_2.
\end{array}\end{displaymath} (20)

Now we have all we need (the operators for tensorial structure and their submatrix elements) for obtaining the value of a matrix element of these operators for any number of open shells in bra and ket functions. This lets us exploit all advantages of the approach by P2 [4].

The spin-spin and spin-other-orbit operators themselves generally contain tensorial structure of several different types. Therefore the expression (1) must be used separately for each possible tensorial structure for performing spin-angular integrations according to P2 [4]. Each type of tensorial structure is associated with a different type of recoupling matrix $R\left( \lambda _i,\lambda _j,\lambda _i^{\prime },\lambda _j^{\prime },
\Lambda ^{bra},\Lambda ^{ket},\Gamma \right) $ and with different matrix elements of standard tensorial quantities $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 one-particle operator is treated in a similar manner. Now the expressions are much simpler and therefore we do not present them here, for brevity.


next up previous
Next: Description of the Libraries Up: Notations and Methodology of Previous: Matrix Elements Between Complex
2001-12-07