Essentials of Computational Chemistry

238 7 INCLUDING ELECTRON CORRELATION IN MO THEORY
7.7.1 Scaling Correlation Energies
The premise behind correlation scaling is particularly simple. Because of basis-set limitations
and approximations in the correlation treatment, one is very rarely able to compute the full
correlation energy. However, with a given choice of basis set and level of theory, the fraction
that is calculated is often quite consistent over a fairly large range of structure. Thus, we
might define an improved electronic energy as
ESAC−e.c.m. = EHF + Ee.c.m. − EHF
A
(7.60)
where ‘e.c.m.’ is a particular electron correlation method, A is an empirical scale factor
typically less than one, and thus all of the correlation energy, computed as the difference
between Eecm and EHF, is scaled by the constant factor of A −1 . SAC emphasizes this
‘scaling all correlation’ energy assumption. Note that the difference between SAC and the
extrapolation schemes of section 7.6.1 is that the latter extrapolate the correlation energy
associated with a given electronic structure model to an infinite basis set, but SAC attempts
to estimate all of the correlation energy.
As first proposed by Gordon and Truhlar (1986), typically one would go about selecting A
by comparison to known experimental data in a system of interest and/or systems related to
it. For example, if the subject of interest is the PES for the reaction of the hydroxyl radical
with ethyl chloride, and if the overall energies of reaction are known for the abstraction of the
α and β hydrogen atoms (to make water and the corresponding alkyl radicals), then A would
be selected for a given electron correlation method (say, MP2) in order to make ESAC−MP2
agree with experiment as closely as possible for those particular data points. This same value
of A would then be used for any point on the PES. Of course, the more experimental details
that can be included in the choice of A, the better the parameterization (and the better able
one is to judge the utility of Eq. (7.60) by examination of the errors in a one-parameter fit).
Note that one particularly attractive feature of Eq. (7.60) is that if the particular electron
correlation method has available analytic derivatives, so too must ESAC−e.c.m., since derivatives
for the latter will be simply determined as appropriately scaled sums of the e.c.m. and
HF derivatives. Geometry optimization, and indeed the entire calculation, can essentially be
carried out for exactly the cost of the e.c.m.
While one might imagine that values of A might best be determined individually within
any given system, Siegbahn and co-workers have examined a large number of primarily
small inorganic systems and suggested that, for the modified coupled-pair functional (MCPF)
treatment of correlation (which is analogous in spirit to coupled cluster) with a polarized
double-ζ basis set, a value of 0.80 has broad applicability, and they name this choice PCI-
80 (Siegbahn, Blomberg, and Svensson 1994; Blomberg and Siegbahn 1998). A summary
of the utility of this level of theory for inorganic systems including comparison to density
functional theory (DFT) can be found in Table 8.2.
Gordon and Truhlar (1986) have emphasized that variations on the theme of Eq. (7.60) can
be useful in different circumstances. For instance, one might imagine carrying out multireference
calculations and assuming two different scale factors, one applying to the non-dynamical