Essentials of Computational Chemistry

148 5 SEMIEMPIRICAL IMPLEMENTATIONS OF MO THEORY
heavier elements. However, in the latter case, the difference is essentially entirely within
the subset of hypervalent molecules included in the test set, e.g., PBr5, IF7, etc. Over the
‘normal’ subset of molecules containing heavy atoms, the performance of AM1 and PM3 is
essentially equivalent. Analysis of the errors in predicted heats of formation suggests that
they are essentially random, i.e., they reflect the ‘noise’ introduced into the Schrödinger
equation by the NDDO approximations and cannot be corrected for in a systematic fashion
without changing the theory. This random noise can be problematic when the goal is to
determine the relative energy differences between two or more isomers (conformational or
otherwise), since one cannot be as confident that errors will cancel as is the case for more
complete quantum mechanical methods.
Errors for charged and open-shell species tend to be somewhat higher than the corresponding
errors for closed-shell neutrals. This may be at least in part due to the greater
difficulty in measuring accurate experimental data for some of these species, but some problems
with the theory are equally likely. For instance, the more loosely held electrons of an
anion are constrained to occupy the same STO basis functions as those used for uncharged
species, so anions are generally predicted to be anomalously high in energy. Radicals are
systematically predicted to be too stable (the mean signed error over the radical test set in
Table 5.2 is only very slightly smaller than the mean unsigned error) meaning that bond
dissociation energies are usually predicted to be too low. Note that for the prediction of
radicals all NDDO methods were originally parameterized with a so-called ‘half-electron
RHF method’, where the formalism of the closed-shell HF equations is used even though
the molecule is open-shell (Dewar, Hashmall, and Venier 1968). Thus, while use of so-called
‘unrestricted Hartree–Fock (UHF)’ technology (see Section 6.3.3) is technically permitted
for radicals in semiempirical theory, it tends to lead to unrealistically low energies and is thus
less generally useful for thermochemical prediction (Pachkovski and Thiel 1996). Finally,
PM3 exhibits a large, non-systematic error in the prediction of proton affinities; AM1 is
more successful for the prediction of these quantities.
For the particular goal of computing accurate heats of formation, Repasky, Chandrasekhar,
and Jorgensen (2002) have suggested a modification to the original approach taken by Dewar
as outlined in Section 5.4.2. Instead of treating a molecule as being composed of atoms as
its fundamental building blocks, they propose 61 common bond and group equivalents that
may instead be considered as small transferable elements. Each such bond or group is then
assigned its own characteristic heat of formation, and a molecular heat of formation is
derived from adding the difference between the molecular electronic energy and the sum of
the fragment electronic energies to the sum of the bond and group heats of formation. In the
form of a general equation we have
H o
N
f,298 (molecule) = E(molecule) − E(fragmenti )
i=1
N
+ H o
f,298 (fragmenti )
(5.17)
i=1
where E is the semiempirical electronic energy. In the original Dewar protocol, the fragments
were atoms and N was the total number of atoms in the molecule. In the bond/group