Cancellation of errors

Calculations are frequently done on the basis of isodesmic reactions in order to provide for maximum cancellation of errors in the total energies (see Section 1.3). The "experimental AH of the process can be obtained from the tabulated AHf values of the reactants and products. Table 4.2 compares the errors in AHf for some isodesmic reactions with those for the corresponding atomization reactions for G2 calculations on some... 

With these results, we see an example of a phenomenon that occurs from time to time a less accurate model chemistry will produce a better answer than more accurate ones. In our case, fortuitous cancellation of errors at the HF/STO-3G levels leads to the correct identification of the planar conformation as a transition state. However, a more accurate model chemistry is needed to properly study this system. ... 

In this reaction, there are twelve single bonds and one (C-O) double bond in both the reactants and products. Because of this conservation of the total number and types of bonds, very good results can be obtained relatively inexpensively for isodesmic reactions due to the cancellation of errors on the two sides of the reaction. In other words, comparing very similar systems enables us to take maximum advantage of cancellation of error. 

The Hartree-Fock values range firom good to quite poor. For the first reaction, cancellation of errors allows Hartree-Fock theory to predict a good value for AH (it overestimates the energies for both ethane and acetone, and underestimates the one for acetaldehyde). 

AMI benefits from the same cancellation of errors for the first reaaion as Hartree-Fock theory. However, it performs even more poorly for the other two reactions. ... 

It can be seen that both of the CPA s reproduce the concentration dependence of Umix quite well, even though it is not a simple parabola. The fact that a cancellation of errors causes the SCF-KKR-CPA results to be quite near the LSMS ones may be one of the reasons that results obtained over the years with that approach have been quite acceptable. After the inclusion of the Coulomb bonding, the CPA-LSMS gives a Umix that is surprisingly good considering other aspects of the approximation that Will be illustrated below. 

The occurrence of such cancellation of errors implies that, in the investigation of an energy difference, one cannot consider a good agreement between a theoretical value and experiment as an... 

It is clear that, in the long run, we cannot rely on a more or less fortuitous cancellation of errors, the nature of which we do not know about in detail. The theory may be in a dilemma, but the only way out is probably to overcome our desire to get quick results and to start investigating the correlation effects in full detail. 

A weakness of the development in the literature up to now has been that too much effort has been concentrated on the helium problem, whereas more complicated systems have been only scarce-ly treated. The reason is obvious it is much easier to test a new method for treating correlation on the ground state of helium, and if the method fails on this simple system, it will certainly not work on a more complicated system either. In treating energy differences in many-electron systems, simple methods will often produce results in excellent agreement with experiment owing to a fortuitous cancellation of errors, but a test on helium will then often reveal the faults of the approach. Even in the future, one can therefore expect that the helium problem will be paid a great deal of interest. 

The authors claim that apparent exceptions, where IOS is more accurate than CS are presumably due to a fortuitous cancellation of errors . The relative mistake in cross-section calculations was defined as... 

Predictive kinetics requires accuracies that are an order of magnitude more precise. There are many examples that predict overall kinetics quite accurately. This is then due to a fortuitous cancellation of errors that needs to be understood well for each case. 

Ab initio calculations usually begin with a solution of the Hartree-Fock equations, which assumes the electronic wavefunction can be written as a single determinant of molecular orbitals. The orbitals are described in terms of a basis set of atomic functions and the reliability of the calculation depends on the quality of the basis set being used. Basis sets have been developed over the years to produce reliable results with a minimum of computational cost. For example, double zeta valence basis sets such as 3-21G [15] 4-31G [16] and 6-31G [17] describe each atom in the molecule with a single core Is function and two functions for the valence s and p functions. Such basis sets are commonly used, as there appears to be a cancellation of errors, which fortuitously allows them to predict quite accurate results. 

However, the values in Tables 26-30 are directly calculated BDEs, according to equation 1. Taking only those molecules for which there are reliable experimental information (Table 26), and excluding the problematic (S 2) cases (CCH-, COOH-, NCO-, CN-, NC-, CHCH2- and N3 ) and the decomposed radical (CH3CO2 ), the average MP2 error for CH3—Y is 2.0 kcalmol-1 (21 cases) and for MP4 is 3.8 kcal mol-1 (19 cases). It thus seems that the particular choice of basis set and level of calculation used here gives the best cancellation of errors for the MP2 method. 

While the PM3-SM4 model does appear to slightly underestimate the polarity of the enol component, there is some cancellation of errors upon considering the differential transfer free energies between cyclohexane and water. As noted above, experiment indicates that the differential free energy of transfer of the dione and the enol is 3.1 kcal/mol the PM3-SM4 model predicts this value to be 2.8 kcal/mol, in excellent quantitative agreement. AM1-SM4 is less satisfactory in this regard, predicting only 1.9 kcal/mol. 

There was a thermodynamic preference for the reaction to take place at the terminal alkene carbon, which favors the yield of linear aldehyde, but the TS to linear aldehyde path was higher than the TS for the branched aldehyde path. Regioselectivity was evaluated from the products relative stability, i.e. considering that the reaction is under thermodynamic rather than under kinetic control. The linear to branched ratio (l b) of 94 6 was in excellent agreement with the ratio 95 5 reported for PPh3 [25], However, this nice coincidence must be viewed cautiously because the model is simple, reaction paths were partially considered, so a subtle cancellation of errors may have been made. 

Let us make two more observations about the convergence of the coupled-cluster hierarchy. First, it converges faster for HF than for N2, reflecting the more complicated electronic structure of the multiple bond in N2. Second, the cancellation of errors in the calculated AEs becomes less pronounced as we move up in the coupled-cluster hierarchy. The cancellation diminishes since the coupled-cluster expansions of the atoms... 

Fortuitous cancellation of errors is less likely and inherent tendencies of the method under examination may be revealed. 

The pole strength is a useful diagnostic criterion of problematic cases. Close agreement with experiment in the presence of a pole strength that is less than 0.80 is likely to be the result of a fortuitous cancellation of errors. 

Because of cancellation of errors in reactions such as (4.5), reasonable results are often obtained, even at quite simple levels of theory. However, it has been found [21, 34] that larger errors may occur with unrestricted methods if there is a significant difference between the degrees of spin contamination for the two radicals in the reaction. 

It is often assumed that there will be substantial cancellation of errors associated with the calculation of stabilization energies via reactions such as (6.2). However, this is not always the case. In particular, it has recently been shown [21, 34] that stabilization energies calculated for the cyanomethyl and cyanovinyl radicals show large variation with level of theory. For these situations, methods such as UMP2 perform very poorly because errors associated with spin contamination in the reactant and product radicals are very different and do not cancel. 

Table 6.3 lists a few H/D isotope effects (kH/kD) for some of the examples listed in Table 6.2. It is noted with some regret that the errors in the isotope effects (kH/kD) calculated by ICVT/ p,OMT are about the same as the errors reported for kH in Table 6.1. One might have hoped to see some cancellation of error when calculating isotope effects.

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