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the results were said to range from excellent for dihydrogen complexes to very poor for complexes [55].

The parameterization of SE methods is supposed to simulate, amongst other effects, electron correlation, so it might seem pointless to introduce electron correlation explicitly, by the Møller–Plesset method or by configuration interaction (section 5.4). However, the parameterization of these SE methods is done using ordinary stable molecules. Surprisingly, MNDO, AM1, PM3 (and presumably SAM1) also reproduce reasonably well the energies and geometries of reactive intermediates like carbocations, carbanions and carbenes. However, the parameterization is unlikely to be as reliable for transition states as for ground states, so activation energies are expected to be less accurate than reaction energies. A method that explicitly calculates electron correlation might improve calculated activation energies. In the SE field the standard program for this is MNDOC (MNDO with correlation), developed by Thiel and coworkers [56]. MNDOC is said to perform better than MNDO (and presumably better than the other MNDO-related methods) in calculating activation energies and electronic excitation energies [39]. For more on the accuracy of MNDO and MNDOC see sections 6.3.1 (geometries) and 6.3.2.2 (activation energies).

6.3 APPLICATIONS OF SE METHODS

A good, brief overview of the performance of MNDO, AM1, and PM3 is given by Levine [57], Hehre has compiled an extremely useful book comparing AM1 with MM (chapter 3), ab initio (chapter 5) and DFT (chapter 7) methods for calculating geometries and other properties [58], and an extensive collection of AM1 and PM3 geometries is to be found in Stewart’s second PM3 paper [44].

6.3.1 Geometries

Many of the general remarks on molecular geometries in section 5.5.1, preceding the discussion of results of specifically ab initio calculations, apply also to SE calculations. Geometry optimizations of large biomolecules like proteins and nucleic acids, which a few years ago were limited to MM, can now be done routinely [59] with SE methods on inexpensive personal computers with the program MOZYME [60], which uses localized

used. Figure 6.2 gives bond lengths and angles calculated by these three methods and experimental bond lengths and angles, for the same 20 molecules as in Fig. 5.23. The geometries shown in Fig. 6.2 are analyzed in Table 6.1, and Table 6.2 provides information on dihedral angles for the same eight molecules as in Table 5.8. Fig. 6.2 corresponds exactly to Fig. 5.23, Table 6.1 to Table 5.7, and Table 6.2 to Table 5.8.

This survey suggests that: AM1 and PM3 give quite good geometries (although dihedral angles, below, show quite significant errors): bond lengths are mostly within 0.02 Å of experimental (although the AM1 C-S bonds are about 0.06 Å too short), and

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Both AM1 and PM3 give quite good bond angles (largest error ca. 4°, except for HOF for which the AM1 error is 7.1°).

AM1 tends to overestimate dihedrals (10+, 0–), while PM3 may do so to a lesser extent (7+, 3–). PM3 breaks down for HOOH (calculated 180°, experimental 119.1°, and does poorly for (calculated 57°, experimental 73°). Omitting the case of HOOH, the mean dihedral angle errors for AM 1 and PM3 are 5° and 4.5°; however, the variation here is from 1 ° to 11 ° for AM 1 and from –1 ° to –16° for PM3 (although not wildly out of line with the AM1, PM3 or MP2 calculations, the reported experimental dihedral of 58.4° is suspect; see section 5.5.1).

The accuracy of AM1 and PM3 then is quite good for bond lengths and angles, but fairly approximate for dihedrals. The largest error (Table 6.1) in bond lengths is 0.065 Å (AM1 for MeSH) and in bond angles 7.1° (AM1 for HOF). The largest error in dihedrals (Table 6.2), omitting the PM3 result for HOOH, is 16° (PM3 for

From Fig. 6.2 and Table 6.1, the mean error in 39(13 + 8 + 9 + 9) bond lengths is ca. 0.01–0.03 Å for the AM1 and PM3 methods, with PM3 being somewhat better except for O–H and O–S. The mean error in 18 bond angles is ca. 2° for both AM1 and PM3. From Table 6.2, the mean dihedral angle error for 9 dihedrals for AM1 and PM3 (omitting the case of HOOH, where PM3 simply fails) is ca. 5°; if we include HOOH, the mean errors for AM1 and PM3 are 6° and 10°, respectively.

Schröder and Thiel have compared MNDO (section 6.2.5.3) and MNDOC (section 6.2.5.7) with ab initio calculations for the study of the geometries and energies of 47 transition states [62], AM1 and PM3 calculations should give somewhat better results than MNDO for these systems, since these two methods are essentially improved versions of MNDO. The general impression is that the SE and ab initio transition states are qualitatively similar in most cases, with MNDOC geometries being sometimes a bit better. The SE and ab initio geometries were in most cases fairly similar, so that as far as geometry goes one would draw the same qualitative conclusions.

Semiempirical and ab initio geometries are compared further in Fig. 6.3, which presents results for four reactions, the same as for the ab initio calculations summarized in Fig. 5.21. As expected from the results of Fig. 6.2, the SE geometries of the reactants and products (the energy minima) are quite good (taking the MP2/6-31G* results as our standard). The SE transition state geometries, however, are also surprisingly good: with only small differences between the AM1 and PM3 results, in all four cases the SE transition states resemble the ab initio ones so closely that qualitative conclusions based on geometry would be the same whether drawn from the AM1 or PM3, or from the MP2/6-31G* calculations. The largest bond length error (if we accept the MP2 geometries as accurate) is about 0.09 Å (for the transition state, 1.897–1.803), and the largest angle error is 9° (for the HNC transition state, 72.8°–63.9°; most of the angle errors are less than 3°).

These results, together with those of Schröder and Thiel [62] indicate that SE geometries are usually quite good, even for transition states. Exceptions might be expected for

predict this species to have a hydrogen-bridged structure (Fig. 5.17). SE energies are considered in section 6.3.2.