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better than those from RHF calculations, at least in many cases. An extensive comparison of AM1 with ab initio and density functional methods for calculating geometries and relative energies is given in Hehre’s book [58]. Consistently good calculated reaction energies and especially activation energies require correlated ab initio methods (sections 5.4.2 and 5.5.2) or DFT methods (chapter 7). It is interesting that AM1 and PM3, which were parameterized mainly to give good energies (heats of formation) actually provide quite good geometries but energies of only modest quality.

6.3.3 Frequencies

The general remarks and the theory concerning frequencies in section 5.5.3, apply to SE frequencies too, but the zero-point energies are usually not needed, since the SE energy is normally not adjusted by adding the ZPE (section 6.2.5.2). As with ab initio calculations, SE frequencies are used to characterize a species as a minimum or a transition state (or a higher-order saddle point), and to get an idea of what the IR spectrum looks like. As with ab initio frequencies too, in SE methods the wavenumbers (frequencies) of vibrations are calculated from a mass-weighted second-derivative matrix (a hessian) and intensities are calculated from the changes in dipole moment accompanying the vibrations. Like their ab initio counterparts, SE frequencies are higher than the experimental ones; presumably this is at least partly due to the harmonic approximation, as was discussed in section 5.5.3.

Correction factors improve the fit between SE calculated and experimentally measured spectra, but the agreement does not become as good as does the fit of corrected

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ab initio to experimental spectra. This is because deviations from experiment are less systematic for SE than for ab initio methods (a characteristic that has been noted for errors in SE energies [68]). For AM1 calculations, correction factors of 0.9235 [69] and 0.9532 [70], and for PM3, factors of 0.9451 [69] and 0.9761 [70], have been recommended. A factor of 0.86 has been recommended for SAM1 for non-H stretches [71]. However, the variation of the correction factor with the kind of frequency is bigger for SE than for ab initio calculations; for example, for correcting carbonyl stretching frequencies, examination of a few molecules indicated (author’s work) that (at least for C, H, O compounds) correction factors of 0.83 (AM1) and 0.86 (PM3) give a much better fit to experiment. In Table 6.4 the factors of Ref. [69] were used. The after-correction deviations from experiment are considerably larger for AM1 and PM3 than for even moderate-level ab initio calculations: about 35% of the SE frequencies deviated by more than 10% from experiment, compared with 21% for HF/3–21G and 10% for HF/6-31G* [70].

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The calculated intensities of SE vibrations are much more approximate than those for ab initio vibrations (the latter are typically within 30% of the experimental intensity at the MP2 level), which is somewhat surprising, since SE (AM1 and PM3) dipole moments, from the vibrational changes of which intensities are calculated, are fairly accurate (section 6.3.4). Note that unlike the case with UV spectra, IR intensities are rarely actually measured; rather, one usually simply visually classifies a band as strong, medium, etc., by comparison with the strongest band in the spectrum. An idea of the reliability of SE frequencies and intensities is given by the IR spectra in Figs 6.5–6.8, which compare experimental spectra with semiempirical (AM1 and PM3) and ab initio (MP2/6-31G*) spectra, for the same four compounds (acetone, benzene, dichloromethane, methanol) shown in Figs 5.33–5.36. The SE spectra are based on the data in Table 6.4; the experimental and MP2 spectra are the ones used in Figs 5.33–5.36. Recall that the MP2/6-31G* IRs of these four compounds do not differ dramatically from the spectra calculated at the HF/3-21G(*) and HF/6-31G* levels. The indication is that SE IR spectra tend to overemphasize the intensities of the strongest bands at the expense of the weaker, with the result that the spectra (after one discards the weakest bands) show fewer bands than experimental or ab initio calculated spectra. Thus if one ignores bands with less than 2% of the intensity of the strongest band (the carbonyl stretch), the AM1 and PM3 spectra of acetone show only carbonyl stretch, while with the 2% cutoff the MP2/6-31G* spectrum has 11 bands and matches the experimental spectrum tolerably well (Fig. 6.5). The semiempirical IRs of benzene (Fig. 6.6) are not bad, but those of dichloromethane (Fig. 6.7) and methanol Fig. 6.8) are also significantly sparser than the experimental and MP2 spectra. Of these four compounds, the SE methods do the best job for benzene, which has only two kind of

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