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Full CI Algorithm

It is easy to vectorize the formation of $\sigma _1$ since each element $\sigma_1(I_{\alpha}, I_{\beta})$ can be written as independent of $I_{\alpha}$ apart from multiplication by a factor $C(I_{\alpha}, J_{\beta})$. The vectorized algorithm for the evaluation of $\sigma _1$, adapted from reference [16], appears in Figure 5. An analogous algorithm can be used to obtain $\sigma_2$. However, we can also obtain $\sigma_2$ for Ms = 0 states by

 \begin{displaymath}
\sigma_2(I_{\alpha}, I_{\beta}) = (-1)^{S} \sigma_1(I_{\beta}, I_{\alpha})
\end{displaymath} (tex2html_deferredtex2html_deferred6.tex2html_deferred29 _2(I_, I_) = (-1)^S _1(I_, I_) )

as proven by Olsen   and co-workers [16].
  
Figure: Vectorized Algorithm for $\sigma _1$.
\begin{figure}\par\begin{verse}
loop over $I_{\beta}$\space \\
\hspace{0.4in} S...
..._{\alpha}$\space \\
end loop over $I_{\beta}$\space \\
\end{verse}\end{figure}

A fairly straightforward algorithm for evaluating $\sigma _3$ is presented in Figure 6. The vectorized algorithm for the evaluation of $\sigma _3$ is presented in Figure 7. This algorithm makes use of a gather and scatter operation to avoid indirect addressing.


  
Figure: Simple Algorithm for $\sigma _3$.
\begin{figure}\par\begin{verse}
loop over $I_{\alpha}$\space \\
\hspace{0.4in} ...
...gn}(kl) (ij\vert kl) C(J_{\alpha}, J_{\beta})$\space \\
\end{verse}\end{figure}


  
Figure: Vectorized Algorithm for $\sigma _3$.
\begin{figure}\par\begin{verse}
loop over $kl$\space \\
\hspace{0.4in} set up $...
...nd loop over $I_{\beta}$\space \\
end loop over $kl$\end{verse}\par\end{figure}

For Ms=0, an improvement to the $\sigma _3$ algorithm can be made by utilizing an identity similar to equation (6.31). The ijklth component of $\sigma _3$ is related to the klijth component via

\begin{displaymath}\sigma_3^{ijkl}(I_{\alpha}, I_{\beta}) =
(-1)^{S} \sigma_1^{klij}(I_{\beta}, I_{\alpha})
\end{displaymath} (tex2html_deferredtex2html_deferred6.tex2html_deferred30_3^ijkl(I_, I_) = (-1)^S _1^klij(I_, I_) )

 


next up previous contents index
Next: Bibliography Up: Determinant-Based CI Previous: Olsen's Full CI Equations
C. David Sherrill
2000-04-18