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

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

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

$$D_{n N_\alpha N_\beta} = \left( \begin{array}{c} n \\ N... ...eft( \begin{array}{c} n \\ N_\beta \\ \end{array} \right)$$

or, in a form closer to equation 4.15,

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

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	$16.5 \times 10^6$	$751 \times 10^6$	$20.3 \times 10^9$	$353 \times 10^9$
	( $4.56 \times 10^6$)	( $172 \times 10^6$)	( $4.04 \times 10^9$)	( $62.5 \times 10^9$)