Density functional models B3LYP

Figure 5. Normal modes for vibration of tetrahedral [Cr04] (chromate). There are four distinct vibrational frequencies, including one doubly-degenerate vibration (E symmetry) and two triply-degenerate vibrations (F2 symmetry), for a total of nine vibrational modes. Arrows show the characteristic motions of each atom during vibration, and the length of each arrow is proportional to the magnitude of atomic motion. Only F2 modes involve motion of the central chromium atom, and as a result their vibrational frequencies are affected by Cr-isotope substitution. The normal modes shown here were calculated with an ab initio quantum mechanical model, using hybrid Hartree-Fock/Density Functional Theory (B3LYP) and the 6-31G(d) basis set—other ab initio and empirical force-field models give very similar results.

Recently, density functional calculations (B3LYP) on the thermal rearrangement of tris(silyl)hydroxylamines to silylamino disiloxane for model compounds concluded that the insertion of a silyl group into the N—O bond is energetically favoured if it occurs from the nitrogen atom. ... 

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. 

B3LYP density functional models provide somewhat better bond length results than the other density functional models, generally very close to experimental distances and to those from MP2 calculations. As with the other density functional models, the errors are largest where one (or two) second-row elements are involved. This is apparent from Figure 5-4, which compares B3LYP/6-311+G and experimental bond distances. 

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. 

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. 

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... 

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. 

It was this observation which gave rise to so-called hybrid derrsity ftmctional models, such as the B3LYP model. Here, the Hartree-Fock exchange energy is added to the exchange energy from a partictrlar density functional model with one or more adjustable parameters. 

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 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. 

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. 

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. 

C=C stretching frequencies experimentally range from 1570 cm in cyclobutene to 1872 cm in tetrafluoroethylene (see appropriate tables in Appendix A7). All levels of calculation reproduce the basic trend in frequencies but, on the basis of mean absolute errors, show widely different performance (Table 7-5). Local density and MP2 models with the 6-311+G basis set perform best and semi-empirical models and density functional models (except the B3LYP model) with the 6-3IG basis set perform worst. Hartree-Fock models with the 3-2IG and larger basis sets also turn in good performance. 

Individual activation energies from BP, BLYP, EDFl and B3LYP density functional models are similar (and different from those of Hartree-Fock and local density models). They are both smaller and larger than standard values, but typically deviate by only a few kcal/mol. The most conspicuous exception is for Diels-Alder cycloaddition of cyclopentadiene and ethylene. Density functional models show activation energies around 20 kcaPmol, consistent with the experimental estimate for the reaction but significantly larger than the 9 kcal/mol value obtained from MP2/6-311+G calculations. Overall, density functional models appear to provide an acceptable account of activation energies, and are recommended for use. Results from 6-3IG and 6-311+G basis sets are very similar, and it is difficult to justify use of the latter. 

All density functional models exhibit similar behavior with regard to dipole moments in diatomic and small polyatomic molecules. Figures 10-6 (EDFl) and 10-8 (B3LYP) show clearly that, except for highly polar (ionic) molecules, limiting (6-311+G basis set) dipole moments are usually (but not always) larger than experimental values. 

Individual errors are typically quite small (on the order of a few tenths of a debye at most), and even highly polar and ionic molecules are reasonably well described. Comparison of results from 6-3IG and 6-311+G density functional models (Figure 10-5 vs. 10-6 for the EDFl model and Figure 10-7 vs. 10-8 for the B3LYP model) clearly reveals that the smaller basis set is not as effective, in particular with regard to dipole moments in highly polar and ionic molecules. Here, the models underestimate the experimental dipole moments, sometimes by 1 debye or more. 

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