Local density models

Just to remind you, the electron density and therefore the exchange potential are both scalar fields they vary depending on the position in space r. We often refer to models that make use of such exchange potentials as local density models. The disagreement between Slater s and Dirac s numerical coefficients was quickly resolved, and authors began to write the exchange potential as... 

Dunlap BI, Connolly JWD, Sabin JR (1979) On first row diatomic molecules and local density models. J Chem Phys 71 4993... 

Three types of exchange/correlation functionals are presently in use (i) functionals based on the local spin density approximation, (ii) functionals based on the generalized gradient approximation, and (iii) functionals which employ the exact Hartree-Fock exchange as a component. The first of these are referred to as local density models, while the second two are collectively referred to as non-local models or alternatively as gradient-corrected models. 

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. 

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. 

Results from local density models and BP, BLYP and EDF 1 density functional models are, broadly speaking, comparable to those from 6-3IG models, consistent with similarity in mean absolute errors. As with bond length comparisons, BLYP models stand out as inferior to the other non-local models. Both B3LYP/6-31G and MP2/6-31G models provide superior results, and either would appear to be a suitable choice where improved quality is required. 

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. 

All density functional models (including the local density model) and the MP2/6-31G model perform admirably in describing the structures of these compounds. In terms of mean absolute errors, the local density model fares best and the BLYP model fares worse. The former observation is consistent with the favorable performance of Hartree-Fock models for these systems and of the previously noted parallels in structural results for Hartree-Fock and local density models. Figures 5-32 to 5-37 provide an overview. 

Interestingly, geometries from Hartree-Fock and local density models with the 6-3IG basis set are markedly different, in contrast to the similarity normally seen in dealing with main-gronp componnds. 

Density functional models provide a much better account. The local density model does the poorest and BP and B3LYP models do the best, but the differences are not great. As with metal-carbon (carbon monoxide) lengths, bond distances from all-electron 6-3IG calculations are usually (but not always) shorter than those obtained... 

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. 

Comparative data for a few particularly interesting systems is provided in Table 5-15. STO-3G, 3-21G and 6-3IG Hartree-Fock models, local density models, BP, BLYP, EDFl and B3LYP density functional models all with the 6-3IG basis set, the MP2/6-31G model and MNDO,AMl andPM3 semi-empirical models have been examined. 

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. 

There is a very wide variation in the quality of results from the different models. MNDO and AMI semi-empirical models, the ST0-3G model and both local density models are completely unsatisfactory. The 3-2IG model, all density functional models with the 6-3IG basis set and the PM3 model fare better, while 6-3IG ... 

Local density models also provide a poor account of homolytic bond dissociation energies. The direction of the errors is the opposite as noted for Hartree-Fock models (reaction energies are too large), but the magnitudes of the errors are comparable. In fact, the average of Hartree-Fock and local density homolytic bond dissociation energies is typically quite close to the experimental energy. ... 

Semi-empirical models do not provide an adequate description of bond dissociation energies and should not be used for this purpose. Errors are not systematic, as was the case for Hartree-Fock models (bond energies too small) and local density models (bond energies too large). Rather, significant errors in both directions are observed. 

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. 

The excellent performance of the local density model with the 6-311+G basis set is unexpected and not easily explained. The mean absolute error is only 3 kcal/mol, a factor of three lower than any other model examined, and the largest single deviation is only 7 kcal/mol (out of 400 kcal/mol). 

This means that diffuse basis functions will be required, just as they were for absolute acidity comparisons. Note, however, that in the case of absolute acidities, none of the models, except the local density model, gave satisfactory results even with the 6-311+G basis set. 

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. 

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. 

Local density models yield bond separation energies of similar quality to those from corresponding (same basis set) Hartree-Fock models. Bond separation energies for isobutane and for trimethylamine, which were underestimated with Hartree-Fock models, are now well described. However, local density models do an even poorer job than Hartree-Fock models with benzene and with small-ring compounds. 

BP, BLYP, EDFl and B3LYP density functional models all lead to significant improvements over both Hartree-Fock and local density models, at least in terms of mean absolute deviations. While most reactions are better described, there are exceptions. Most notable among these is the bond separation reaction for tetrachloromethane. All four models show a highly exothermic reaction in contrast with both G3 and experimental results which show a nearly thermoneutral reaction. Similar, but somewhat smaller, effects are seen for isobutane and trimethylamine. As was the case with Hartree-Fock calculations. 

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. 

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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