Essentials of Computational Chemistry

5.1 SEMIEMPIRICAL PHILOSOPHY 133
so that room-temperature predictions can be trusted at least to within an order of magnitude
(and obviously it would be nice to do much better). How then can we ever hope to use a
theory that is intrinsically inaccurate by hundreds or thousands of kilocalories per mole to
make chemically useful predictions? Michael J. S. Dewar, who made many contributions in
the area of semiempirical MO theory, once offered the following analogy to using HF theory
to make chemical predictions: It is like weighing the captain of a ship by first weighing the
ship with the captain on board, then weighing the ship without her, and then taking the
difference – the errors in the individual measurements are likely to utterly swamp the small
difference that is the goal of the measuring.
In practice, as we shall see in Chapter 6, the situation with HF theory is not really as
bad as our above analysis might suggest. Errors from neglecting correlation energy cancel
to a remarkable extent in favorable instances, so that chemically useful interpretations of
HF calculations can be valid. Nevertheless, the intrinsic inaccuracy of ab initio HF theory
suggests that modifications of the theory introduced in order to simplify its formalism may
actually improve on a rigorous adherence to the full mathematics, provided the new ‘approximations’
somehow introduce an accounting for correlation energy. Since this improves
chemical accuracy, at least in intent, we may call it a chemically virtuous approximation.
Most typically, such approximations involve the adoption of a parametric form for some
aspect of the calculation where the parameters involved are chosen so as best to reproduce
experimental data – hence the term ‘semiempirical’.
5.1.2 Analytic Derivatives
If it is computationally demanding to carry out a single electronic structure calculation, how
much more daunting to try to optimize a molecular geometry. As already discussed in detail
in Section 2.4, chemists are usually interested not in arbitrary structures, but in stationary
points on the potential energy surface. In order to find those points efficiently, many of
the optimization algorithms described in Section 2.4 make use of derivatives of the energy
with respect to nuclear motion – when those derivatives are available analytically, instead
of numerically, rates of convergence are typically enhanced.
This is particularly true when the stationary point of interest is a transition-state structure.
Unlike the case with molecular mechanics, the HF energy has no obvious bias for minimumenergy
structures compared to TS structures – one of the most exciting aspects of MO
theory, whether semiempirical or ab initio, is that it provides an energy functional from
which reasonable TS structures may be identified. However, in the early second half of
the twentieth century, it was not at all obvious how to compute analytic derivatives of
the HF energy with respect to nuclear motion. Thus, another motivation for introducing
semiempirical approximations into HF theory was to facilitate the computation of derivatives
so that geometries could be more efficiently optimized. Besides the desire to attack TS
geometries, there were also very practical motivations for geometry optimization. In the
early days of semiempirical parameterization, experimental structural data were about as
widely available as energetic data, and parameterization of semiempirical methods against
both kinds of data would be expected to generate a more robust final model.