Essentials of Computational Chemistry

Property
∗
6.2 BASIS SETS 177
6 5
Q
n −1
T D
Figure 6.4 Use of an extrapolation procedure to estimate the expectation value for some property
at the HF limit. The abscissa is marked off as n −1 in cc-pVnZ notation (see page 162). Note the
sensitivity of the limiting value, which is to say the ordinate intercept, that might be expected based
on the use of different curve-fitting procedures
equation should take, or can any arbitrary curve fitting approach be applied? In general,
the answers to these questions are case-dependent, and the chemist cannot be completely
removed from the calculation.
Note that the cost of the extrapolation procedure outlined above becomes increasingly large
as points for n = 4, 5, and 6 are added. For systems having more than five or six atoms,
these calculations can be staggeringly demanding in terms of computational resources.
A somewhat more common approach is one that does not try explicitly to extrapolate to
the HF limit but uses similar concepts to try to correct for some basis-set incompleteness. The
assumption is made that the effects of ‘orthogonal’ increases in basis set size can be considered
to be additive (a substantial amount of work suggests that this assumption is typically
not too bad, at least for molecular energies), and thus the individual effects can be summed
together to estimate the full-basis-set result. This is best illustrated by example. Consider
HF calculations carried out for the chemical warfare agent VX (C11H26NO2PS, Figure 6.5)
with the following basis sets: 6-31G, 6-31++G, 6-31G(d,p), 6-311G, and 6-311++G(d,p).
With these basis sets, the total number of basis functions for VX are 204, 378, 294, 294,
and 542, respectively.
The additivity assumption can be expressed as
E[HF/6-311++G(d,p)] ≈ E[HF/6-31G]
+{E[HF/6-31G(d,p)] − E[HF/6-31G]}
+{E[HF/6-311G] − E[HF/6-31G]}
+{E[HF/6-31++G] − E[HF/6-31G]} (6.5)