Essentials of Computational Chemistry

7.7 PARAMETERIZED METHODS 243
where (i) represents a component of the G3 energy (actually, there are some rather slight
variations involved with basis sets and frozen-core approximations that increase efficiency),
ESO and ECC are empirically estimated spin-orbit and core-correlation energies, and the
coefficients ci are optimized over the usual G3 thermochemistry test set. One additional
important difference in the use of G3 energy components is that the G3 empirical correction,
which leads to non-size-extensivity, is not included. Thus, MCG3 is size extensive. The
performance of MCG3 is very slightly better than G3 itself, but this accuracy is achieved at
roughly half the cost in terms of computational resources for molecules having many heavy
atoms. Scaling of the G3 components was also reported by Curtiss et al. (2000) and defines
the G3S model. MCG3 and G3S have essentially equivalent accuracy.
The real power in the multi-coefficient models, however, derives from the potential for
the coefficients to make up for more severe approximations in the quantities used for (i) in
Eq. (7.62). At present, Truhlar and co-workers have codified some 20 different multicoefficient
models, some of which they term ‘minimal’, meaning that relatively few terms enter
into analogs of Eq. (7.62), and in particular the optimized coefficients absorb the spin-orbit
and core-correlation terms, so they are not separately estimated. Different models can thus
be chosen for an individual problem based on error tolerance, resource constraints, need
to optimize TS geometries at levels beyond MP2, etc. Moreover, for some of the minimal
models, analytic derivatives are available on a term-by-term basis, meaning that analytic
derivatives for the composite energy can be computed simply as the sum over terms.
A somewhat more chemically based empirical correction scheme is the bond-additivity
correction (BAC) methodology. In the BAC-MP4 approach, for instance, the energy of a
molecule is computed as
E(BAC-MP4) =E[MP4/6-31G(d,p)//HF/6-31G(d,p)]
+
EA−B + ESC + EMR
A,B
(7.63)
where ESC and EMR correct for spin contamination (if any) and multireference character (if
any) and the summation runs over all atom pairs and each ‘bond’ correction is a function of
bond length (the correction goes to zero at infinite bond length) and a set of parameters, one
parameter for each atom and two parameters for each possible pair of atoms. The parameters
themselves are determined by fitting to experimental bond dissociation energies, heats of
formation (corrected for zero-point vibrational energies and thermal contributions), or other
useful thermochemical data. The central assumption of this model, then, is that the error can
be decomposed in an additive fashion over the bonds.
In a study of 110 C1 and C2 molecules composed of C, H, O, and F, the average BAC-
MP4 unsigned error in predicted heat of formation was 2.1 kcal mol −1 (Zachariah et al.
1996). As the MP4 calculation uses a relatively modest basis set size, the BAC procedure is
quite fast by comparison to some of the multilevel methods described above. On the other
hand, as with any method relying on pairwise parameterization, extension to a large number
of atoms requires a great deal of parameterization data, and this is a potential limitation of
the BAC method when applied to systems containing atoms not already parameterized.