Essentials of Computational Chemistry

6.2 BASIS SETS 171
Table 6.1 STO-3G 2sp basis set for oxygen
α2sp c2s c2p
5.0331527 −0.099967 0.155916
1.1695944 0.399513 0.607684
0.3803892 0.700115 0.391957
defined for each type of orbital core through valence. Thus for H and He, there is only a 1s
function. For Li to Ne, there are five functions, 1s, 2s, 2px, 2py, and 2pz. For Na to Ar, 3s,
3px, 3py, and 3pz are added to the second-row set, making a total of nine functions, etc. This
number is the absolute minimum required, and it is certainly nowhere near the infinite basis
set limit. Other minimal basis sets include the MINI sets of Huzinaga and co-workers, which
are named MINI-1, MINI-2, etc., and vary in the number of primitives used for different
kinds of functions.
One way to increase the flexibility of a basis set is to ‘decontract’ it. That is, we might
imagine taking the STO-3G basis set, and instead of constructing each basis function as a
sum of three Gaussians, we could construct two basis functions for each AO, the first being
a contraction of the first two primitive Gaussians, while the second would simply be the
normalized third primitive. This prescription would not double the size of our basis set, since
we would have all the same individual integrals to evaluate as previously, but the size of our
secular equation would be increased. A basis set with two functions for each AO is called a
‘double-ζ ’ basis. Of course, we could decontract further, and treat each primitive as a fullfledged
basis function, in which case we would have a ‘triple-ζ ’ basis, and we could then
decide to add more functions indefinitely creating higher and higher multiple-ζ basis sets.
Modern examples of such basis sets are the cc-pCVDZ, cc-pCVTZ, etc. sets of Dunning
and co-workers, where the acronym stands for ‘correlation-consistent polarized Core and
Valence (Double/Triple/etc.) Zeta’ (Woon and Dunning 1995); correlation consistency and
polarization are described in more detail below.
The advantage of such a scheme is, naturally, that these increasingly large basis sets must
come closer and closer to the HF limit. Let us step back for a moment, however, and consider
the chemical consequences of providing extra basis functions for a given AO. Recall that a
final MO from an HF calculation is a linear combination of all of the basis functions. Indeed,
if we were to examine the 1s core orbital resulting from an HF calculation on atomic oxygen
using the fully uncontracted set of STO-3G Gaussian primitives as a basis (i.e., a triple-ζ
basis), we might well find it to be a linear combination of the 1s functions very similar to
that defining the STO-3G contracted oxygen 1s function. And, if we were to look at the
MOs resulting from an equivalent calculation on, say, formaldehyde (H2C=O), we would
probably find another orbital which we would assign as the oxygen 1s orbital having very
similar AO coefficients. Indeed, we would find this same orbital little changed in almost any
molecule incorporating oxygen we might choose to examine. The reason for this is that core
orbitals are only weakly affected by chemical bonding.
Valence orbitals, on the other hand, can vary widely as a function of chemical bonding.
Atoms bonded to significantly more electronegative elements take on partial positive charge