Table 3: Number of CSF's required for small molecules at several
levels of CI.
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 H
used DZP basis sets, those on H
O and NH
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
The dimension of the full CI
space in determinants (again, ignoring spatial symmetry) is computed simply by
or, in a form closer to equation 4.15,
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 [, 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)