Essentials of Computational Chemistry

160 5 SEMIEMPIRICAL IMPLEMENTATIONS OF MO THEORY
to give high enantioselectivities in the alkyl addition, indicating that either a single one of
the four TS structures is significantly lower in energy, or, if not, the two associated with
one enantiomer are significantly lower than either of the two for the other enantiomer.
To better determine the specific steric and/or electronic influences giving rise to high
observed enantioselectivities, Goldfuss and Houk studied the energies of the four TS structures
in Figure 5.3 for different chiral β-amino alcohols at the PM3 level of theory.
One possible concern in such an approach is the quality of the Zn parameters in PM3,
since experimental data for zinc compounds are considerably more sparse than for more
quotidian organic compounds. Thus, as a first step, Goldfuss and Houk considered the small
complex formed from formaldehyde, dimethylzinc, and unsubstituted β-aminoethanol. They
compared the geometries of the two TS structures predicted at the PM3 level to those
previously obtained by another group at the ab initio HF/3-21G level (note that since
the amino alcohol is not chiral, there are two TS structures, not four); they observed
that agreement was reasonable for the gross shapes of the TS structures, although there
were fairly substantial differences in various bond lengths – up to 0.2 ˚A in Zn–O bonds
and the forming C–C bond. They also compared the relative energies for the two TS
structures at the PM3 level to those previously reported from small, correlated ab initio
calculations. Agreement was at best fair, with PM3 giving an energy difference between
the two structures of 6.8 kcal mol −1 , compared to the prior result of 2.9 kcal mol −1 .
Comparison between PM3 and the previously reported levels of theory is interesting from
a methodological perspective. However, to the extent that there are significant disagreements
between the methods, PM3 is as likely to be the most accurate as any, given the rather low
levels of ab initio theory employed (ab initio theory is discussed in detail in the next two
chapters). Insofar as the size of the chemical problem makes it impractical to seek converged
solutions of the Schrödinger equation, Goldfuss and Houk turned instead to a comparison
of PM3 to available experimental data. In particular, they computed product ratios based on
the assumption that these would reflect a 273 K Boltzmann distribution of corresponding TS
structures (this follows from transition state theory, discussed in Section 15.3, for a reaction
under kinetic control). For the TS energies, they employed the relative PM3 electronic
energies plus zero-point vibrational energies obtained from frequency calculations (see
Section 10.2). Then, for a variety of different chiral β-amino alcohols, they compared
predicted enantiomeric excess, defined as
%ee =|%R − %S| (5.20)
to experimental values obtained under a variety of different conditions. This comparison,
summarized in Table 5.3, shows remarkably good agreement between PM3 and experiment.
It is worth a brief digression to note that, from a theoretical standpoint, it is rather easy to
make predictions in cases where a single product is observed. When experimentalists report
a single product, they typically mean that to within the detection limits of their analysis,
they observe only a single compound – unless special efforts are undertaken, this might
imply no better than 20:1 excess of the observed product over any other possibilities.
At 298 K, this implies that the TS structure of lowest energy lies at least 2 kcal mol −1
below any competing TS structures. Of course, it might be 20 or 200 kcal mol −1 below
competing TS structures – when experiment reports only a single product, there is no way
to quantify this. Thus, even if theory is badly in error, as long as the correct TS structure is
predicted to be lowest by more than 2 kcal mol −1 , there will be ‘perfect’ agreement with