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Size of the CI Space as a Function of Excitation Level


 
 
Table 3: Number of CSF's required for small molecules at several levels of CI.
  CSF's requireda
Molecule CISD CISDT CISDTQ FCI
BH 568 n/a 28 698 132 686
H2O 361 3 203 17 678 256 473
NH3 461 4 029 19 925 137 321
HF 552 6 712 48 963 944 348
H7+ 1 271 24 468 248 149 2 923 933
aData from reference [6] except for H7+ data from
reference [17].
       

We can also see from Table 3 that the number of N-electron basis functions increases dramatically with increasing excitation level. It should be pointed out that while the calculations on BH, HF, and H7+ used DZP basis sets, those on H2O and NH3 used only DZ basis sets. A DZP basis should be considred the minimum adequate basis for a truly meaningful benchmark study, and even then it is desirable to use a high-quality basis such as an Atomic Natural Orbital (ANO)  set. While it is generally possible to perform CISD  calculations on small molecules with a good one-electron basis, the CISDTQ  method is limited to molecules containing very few heavy atoms, due to the extreme number of N-electron basis functions required. Full CI  calculations are of course even more difficult to perform, so that despite their importance as benchmarks, few full CI  energies using flexible one-electron basis sets have been obtained.

The size of the full CI  space in CSF's can be calculated (including spin symmetry but ignoring spatial symmetry) by Weyl's dimension formula  as applied to the Distinct row table (DRT).    If N is the number of electrons, n is the number of orbitals, and S is the total spin, then the dimension of the CI space in CSF's is given by

 \begin{displaymath}
D_{nNS} = \frac{2S+1}{n+1}
\left( \begin{array}{c}
n + 1 ...
...gin{array}{c}
n + 1 \\
N/2 + S + 1 \\
\end{array} \right)
\end{displaymath} (tex2html_deferredtex2html_deferred4.tex2html_deferred15 D_nNS = 2S+1n+1 (
\begin{array}{c}
n + 1 \\
N/2 - S \\
\end{array}
) (
\begin{array}{c}
n + 1 \\
N/2 + S + 1 \\
\end{array}
) )

The dimension of the full CI  space in determinants (again, ignoring spatial symmetry) is computed simply by

 \begin{displaymath}
D_{n N_\alpha N_\beta} =
\left( \begin{array}{c}
n \\
N...
...eft( \begin{array}{c}
n \\
N_\beta \\
\end{array} \right)
\end{displaymath} (tex2html_deferredtex2html_deferred4.tex2html_deferred16 D_n N_N_ = (
\begin{array}{c}
n \\
N_\alpha \\
\end{array}
) (
\begin{array}{c}
n \\
N_\beta \\
\end{array}
) )

or, in a form closer to equation 4.15,

\begin{displaymath}D_{n N S} =
\left( \begin{array}{c}
n \\
N/2 + S \\
\e...
...eft( \begin{array}{c}
n \\
N/2 - S \\
\end{array} \right)
\end{displaymath} (tex2html_deferredtex2html_deferred4.tex2html_deferred17D_n N S = (
\begin{array}{c}
n \\
N/2 + S \\
\end{array}
) (
\begin{array}{c}
n \\
N/2 - S \\
\end{array}
) )

Table 4 shows the dimension of the full CI  space (neglecting spatial symmetry) in determinants and in CSF's. Current full CI algorithms are limited to a few million determinants. Although there have been reports [18,19,20] of larger calculations (including a few billion determinants), the computational expense is (currently) too great for routine calculations of this size.


 
 
Table 4: Dimension of Full CI in Determinants (CSF's in parentheses)
  Number of electrons
Orbitals 6 8 10 12
10 $14.4 \times 10^3$ $44.1 \times 10^3$ $63.5 \times 10^3$ $44.1 \times 10^3$
  ( $4.95\times 10^3$) ( $13.9\times 10^3$) ( $19.4\times 10^3$) ( $13.9\times 10^3$)
         
20 $1.30 \times 10^6$ $23.5 \times 10^6$ $240 \times 10^6$ $1.50 \times 10^9$
  ( $379 \times 10^3$) ( $5.80 \times 10^6$) ( $52.6 \times 10^6$) ( $300 \times 10^6$)
         
30 $16.5 \times 10^6$ $751 \times 10^6$ $20.3 \times 10^9$ $353 \times 10^9$
  ( $4.56 \times 10^6$) ( $172 \times 10^6$) ( $4.04 \times 10^9$) ( $62.5 \times 10^9$)
 


next up previous contents index
Next: The Frozen Core Approximation Up: Reducing the Size of Previous: Energy Contributions of the
C. David Sherrill
2000-04-18