where the sum over C^ is the trace of the first-order wave function coefficients for the natural orbital pair ii, |Si is the absolute value of the spatial overlap between the a and b spin components of the ith MO, and the factor 0.00533 is determined by fitting to the reference data. This empirical correction is size extensive.

(6) For open-shell species the UHF method is used, which in some cases suffers from spin contamination. To correct for this an empirical correction based on the deviation of <S2> from the theoretical value is added for the CBS-4 and CBS-Q methods, A£emp = -0.0092[<S2> - - 1)], where the factor of -0.0092 is derived by fitting.

The use of the smaller basis for the QCISD(T) calculation means that the CBS-Q model is computationally faster than G2(MP2), but nevertheless gives slightly lower errors. A comparison among the four CBS models is shown in Table 5.6 (p. 218).

It should be noted that the G2-1 data set, with two exceptions (SO2 and CO2), only includes data for molecules containing one or two heavy (non-hydrogen) atoms. It is likely that the typical error for a given model to a certain extent depends on the size of the system, i.e. the G2 method is presumably not able to predict the heat of formation of say C60 (if it were computationally feasible) with an accuracy of ~6kJ/mol. Furthermore, the properties included (atomization energies, ionization potentials, electron and proton affinities) all correspond to energy differences between well-separated systems: atomization energies are energy differences between a molecule and isolated atoms, and the other three properties correspond to removal or addition of a single electron or proton. As illustrated in Chapter 11, such energy differences are easier to calculate than between systems containing half broken/formed bond. As with any scheme that has been parameterized on experimental data, it is questionable to assume that the typical accuracy for a selected set of properties will be true in general. A good performance for the G2 data set does not necessarily indicate that the same level of accuracy can be obtained over a wide variety of geometries, for example including transition structures. A modified version of the G2 method, denoted G2Q, involving geometry optimization and frequency calculation at the QCISD/6-311G(d,p) level, has been advocated by Durant and Rohlfing for use with transition structures.59

The G3 and CBS-APNO methods are capable of calculating average atomization energies to within 2-4kJ/mol,but the maximum error for the reference data set is often significantly larger. Since it is difficult to know in advance whether the particular system of interest behaves as the average or the exceptional case, the predicted value must realistically be assumed to have an uncertainty of perhaps 10-20kJ/mol. Part of the reason for the relatively large spread in the errors is the assumption of additivity in basis sets effect, which has little theoretical foundation, although the empirical corrections at least partly absorb some of these errors.

If higher accuracy is desired, for example "sub-chemical" (~0.5kJ/mol) or "spectroscopic accuracy" (~1cm-1, ~0.01kJ/mol), a number of other factors must also be considered:

(1) Correlation of the core and core-valence electrons. This becomes progressively more important as heavier elements are considered.

Table 5.6 Computational levels in the CBS models



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