MP2 model

An alternative approach to improve upon Hartree-Fock models involves including an explicit term to account for the way in which electron motions affect each other. In practice, this account is based on an exacf solution for an idealized system, and is introduced using empirical parameters. As a class, the resulting models are referred to as density functional models. Density functional models have proven to be successful for determination of equilibrium geometries and conformations, and are (nearly) as successful as MP2 models for establishing the thermochemistry of reactions where bonds are broken or formed. Discussion is provided in Section II. 

Calculated equilibrium geometries for hydrogen and main-group hydrides containing one and two heavy (non-hydrogen) atoms are provided in Appendix A5 (Tables A5-1 and A5-10 for molecular mechanics models, A5-2 and A5-11 for Hartree-Fock models, A5-3 and A5-12 for local density models, A5-4 to A5-7 and A5-13 to A5-16 for BP, BLYP, EDFl and B3LYP density functional models, A5-8 and A5-17 for MP2 models and A5-9 and A5-18 for MNDO, AMI and PM3 semi-empirical models). Mean absolute errors in bond lengths are provided in Tables 5-1 and 5-2 for one and two-heavy-atom systems, respectively. 

Although all models considered provide a generally credible account of the experimental structural data, the variation among methods is somewhat greater than previously noted for hydrocarbons. Here, more so than in the previous comparison, it is evident that the best performers are B3LYP and MP2 models. 

As was the case with hydrocarbons, 6-3IG and 6-311+G basis sets lead to similar bond lengths for all density functional models as well as for the MP2 model. This is reflected in the mean absolute errors. It is difficult to justify use of the larger basis set models for routine structure determinations. 

As with hydrocarbons, accurate descriptions of equilibrium structures for molecules with heteroatoms from density functional and MP2 models requires polarization basis sets. As shown in Table A5-20 (Appendix A5), bond distances in these compounds obtained from (EDF 1 and B3LYP) density functional models and from MP2 models... 

Calculated heavy-atom bond distances in molecules with three or more first and/or second-row atoms are tabulated in Appendix A5 molecular mechanics models (Table A5-21), Hartree-Fock models (Table A5-22), local density models (Table A5-23), BP, BLYP, EDFl and B3LYP density functional models (Tables A5-24 to A5-27), MP2 models (Table A5-28), and MNDO, AMI and PM3 semi-empirical models (Table A5-29). Results for STO-3G, 3-21G, 6-31G and 6-311+G basis sets are provided for Hartree-Fock models, but as in previous comparisons, only 6-3IG and 6-311+G basis sets are employed for local density, density functional and MP2 models. 

None of the semi-empirical models perform as well as Hartree-Fock models (except STO-3G), local density models, density functional models or MP2 models. PM3 provides the best overall description, although on the basis of mean absolute errors alone, all three models perform to an acceptable standard. Given the large difference in cost of application, semi-empirical models clearly have a role to play in structure determination. 

Consistent with earlier remarks made for bond length comparisons, little if any improvement results in moving from the 6-3IG to the 6-311+G basis set for Hartree-Fock, local density and density functional models, but significant improvement results for MP2 models. 

Comparative data for heavy-atom bond lengths and skeletal bond angles for molecules incorporating one or more third or fourth-row, main-group elements are provided in Appendix A5 Table A5-39 for Hartree-Fock models with STO-3G, 3-2IG and 6-3IG basis sets. Table A5-40 for the local density model, BP, BLYP, EDFl andB3LYP density functional models and the MP2 model, all with the 6-3IG basis set, and in Table A5-41 for MNDO, AMI and PM3 semi-empirical models. 6-31G, local density, density functional and MP2 calculations have been restricted to molecules with third-row elements only. Also, molecular mechanics models have been excluded from the comparison. A summary of errors in bond distances is provided in Table 5-8. 

The MP2/6-31G model does not perform as well as any of the density functional models. As for Hartree-Fock models, most individual systems are well described but some are very poorly described. This behavior is perhaps not unexpected, as MP2 models are based on the use of Hartree-Fock wavefimctions. This means that a single electronic configuration is assumed to be better than all other configurations, a situation that is probably umeasonable for this class of compounds. 

As with metal-carbon monoxide bonds, the MP2/6-3IG model does not lead to results of the same calibre as those from density functional models (except local density models). The model actually shows the opposite behavior as 6-3IG, in that bond lengths are consistently shorter than experimental values, sometimes significantly so. In view of its poor performance and the considerable cost of MP2 models (relative to density functional models), there seems little reason to employ them for structural investigations on organometallics. 

Triplet methylene is known to be bent with a bond angle of approximately 136°. This is closely reproduced by all Hartree-Fock models (except for STO-3G which yields a bond angle approximately 10° too small), as well as local density models, BP, BLYP, EDFl and B3LYP density functional models and MP2 models. Semi-empirical models also suggest a bent structure, but with an HCH angle which is much too large. 

Triplet oxygen deserves special attention if for no other reason that it is the only common non-closed shell molecule. As expected, the limiting (6-311+G basis set) Hartree-Fock bond distance is too short. The corresponding B3FYP model provides a bond length in nearly exact agreement with experiment, while all other density functional models and especially the MP2 model significantly overestimate the bond distance. 

There are actually very few. Modern optimization techniques practically guarantee location of a minimum energy structure, and only where the initial geometry provided is too symmetric will this not be the outcome. With a few notable exceptions (Hartree-Fock models applied to molecules with transition metals), Hartree-Fock, density functional and MP2 models provide a remarkably good account of equilibrium structure. Semi-empirical quantum chemical models and molecular mechanics models, generally fare well where they have been explicitly parameterized. Only outside the bounds of their parameterization is extra caution warranted. Be on the alert for surprises. While the majority of molecules assume the structures expected of them, some will not. Treat "unexpected" results with skepticism, but be willing to alter preconceived beliefs. 

It is likely that different quantum chemical models will perform differently in each of these situations. Processes which involve net loss or gain of an electron pair are likely to be problematic for Hartree-Fock models, which treat the electrons as essentially independent particles, but less so for density functional models and MP2 models, which attempt to account for electron correlation. Models should fare better for processes in which reactants and products are similar and benefit from cancellation of errors, than those where reactants and products are markedly different. The only exception might be for semi-empirical models, which have been explicitly parameterized to reproduce individual experimental heats of formation, and might not be expected to benefit from error cancellation. 

Calculations have been performed in order to dissect the observed changes in bond dissociation energies between models with 6-31G and 6-311+G basis sets. These are provided in Appendix A6, Tables A6-9 to A6-11 for EDFl, B3LYP and MP2 models, respectively. No significant changes in bond dissociation energies are noted as a result... 

Density functional models and MP2 models show more consistent behavior. With the 6-311+G basis set, calculated basicities are generally very close to experimental values. The corresponding results with the 6-3IG basis set are generally not as good, although the differences are not that great. In terms of mean absolute errors, local density models perform the worst, and B3LYP/6-311+G and MP2/ 6-311+G models perform the best. 

Data are provided in Table 6-10, with the same calculation models previously examined for hydrogenation reactions. As might be expected from the experience with hydrogenation reactions, Hartree-Fock models with 6-3IG and 6-311+G basis sets perform relatively well. In fact, they turn in the lowest mean absolute errors of any of the models examined. The performance of density functional models (excluding local density models) and MP2 models with both 6-3IG and 6-311+G basis sets is not much worse. On the other hand, local density models yield very poor results in all cases showing reactions which are too exothermic. The reason is unclear. Semi-empirical models yield completely unacceptable results, consistent with their performance for hydrogenation reactions. 

Calculated relative energies for a small selection of structural isomers are compared with experimental values and with the results of G3 calculations in Table 6-11. These have been drawn from a much more extensive set of comparisons found in Appendix A6 (Tables A6-24 to A6-31). Mean absolute errors from the full set of comparisons are collected in Table 6-12, and a series of graphical comparisons involving Hartree-Fock, EDF1, B3LYP and MP2 models... 

Better accounts of relative isomer energies are provided by density functional models and by MP2 models. With both 6-3IG and 6-311+G basis sets, BP, EDFl and MP2 models perform best and BLYP models perform worst, although the differences are not great. In terms of mean absolute errors, all models improve upon replacement of the 6-3 IG by the 6-311+G basis set. With some notable exceptions, individual errors also decrease in moving from the 6-31G to 6-311+G basis sets. (A further breakdown of basis set effects is provided in Tables A6-32 to A6-35 in Appendix A6.) The improvements are, however, not great in most cases, and it may be difficult to justify of the extra expense incurred in moving from 6-3IG to the larger basis set. 

The comparison between propyne and allene warrants additional comment. Experimentally, propyne is the more stable by approximately 2 kcal/mol, an observation which is reproduced by Hartree-Fock models but is somewhat exaggerated by MP2 models. Note, however, that all density functional models (including local density models) show the reverse order of isomer stabilities with allene being more stable than propyne. This is another instance where the behavior of B3LYP and MP2 models do not mimic each other. 

Bond separation energies from Hartree-Fock models with STO-3G, 3-2IG, 6-3IG and 6-311+G basis sets, local density models, BP, BLYP, EDFl and B3LYP density functional models and MP2 models all with 6-3IG and 6-311+G basis sets and MNDO, AMI and PM3 semi-empirical models are compared with values based on G3 energies and on experimental thermochemical data in Table 6-13. These have been abstracted from a much larger collection found in Appendix A (Tables A6-36 to A6-43). A summary of mean absolute deviations from G3 values in calculated bond separation energies (based on the full data set) is provided in Table 6-14. 

With the exception of STO-3G and both MP2 models, all models (including semi-empirical models) provide a credible account of relative CH bond energies. In terms of mean absolute error, BP and B3LYP models with the 6-311+G basis set are best and Hartree-Fock 3-21G and 6-3IG models, local density 6-3IG models and semi-empirical models are worst. More careful scrutiny turns up sizeable individual errors which may in part be due to the experimental data. For example, the best of the models appear to converge on a CH bond dissociation for cycloheptatriene which is 35-37 kcal/mol less than that in methane (the reference compound) compared with the experimental estimate of 31 kcal/mol. It is quite possible that the latter is in error. The reason for the poor performance of MP2 models, with individual errors as large as 16 kcal/mol (for cycloheptatriene) is unclear. The reason behind the unexpected good performance of all three semi-empirical models is also unclear. 

With the exception of semi-empirical models, all models provide very good descriptions of relative nitrogen basicities. Even STO-3G performs acceptably compounds are properly ordered and individual errors rarely exceed 1 -2 kcal/mol. One unexpected result is that neither Hartree-Fock nor any of the density functional models improve on moving from the 6-3IG to the 6-311+G basis set (local density models are an exception). Some individual comparisons improve, but mean absolute errors increase significantly. The reason is unclear. The best overall description is provided by MP2 models. Unlike bond separation energy comparisons (see Table 6-11), these show little sensitivity to underlying basis set and results from the MP2/6-3IG model are as good as those from the MP2/6-311+G model. 

Except for very low values (< 600 cm ), frequencies can normally be measured to high precision (< 5 cm ) using infrared or Raman spectroscopy. Similar or better precision is available for frequencies calculated analytically (Hartree-Fock, density functional and semi-empirical models), but somewhat lower precision results where numerical differentiation is required (MP2 models). 

In terms of both mean absolute error (in symmetric stretching frequencies) and of individual frequencies, density functional models perform significantly better than Hartree-Fock models. As with diatomic molecules, local density models appear to provide the best overall account, but the performance of the other models (except for B3LYP models) is not much different. B3LYP models and MP2 models do not appear to fare as well in their descriptions of frequencies in one-heavy-atom hydrides, and the performance of each appears to worsen in moving from the 6-3IG to the 6-311+G basis set. 

While experimental harmonic frequencies are limited, there are sufficient data to suggest that were they used instead of measured frequencies, local density models would fare worse and MP2 models would fare better. For example, mean absolute errors (based on limited data) for the loeal density 6-311+G model is 121 em while that for the MP2/6-311+G model is 49em. ... 

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