Essentials of Computational Chemistry

232 7 INCLUDING ELECTRON CORRELATION IN MO THEORY
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Figure 7.6 Frontier orbitals of a para aryne diradical. In the isoelectronic cases of X = Cand
X = NH + , the energy of orbital φa is very slightly below that of φb, leading to a high degree of
multiconfigurational character in the singlets. When X = C, the two orbitals belong to the b1u and ag
irreps of the D2h point group, respectively, and thus a single excitation from the former to the latter
cannot contribute to the singlet ground state that has overall Ag symmetry; only a double excitation
contributes. When X = NH + , however, both orbitals are of a ′ type symmetry within the Cs point group,
so that a single excitation can (and does) make a large contribution to the singlet ground state that
has overall A ′ symmetry. The latter situation can contribute to instability in estimating the energetic
effects of triples substitutions in ‘(T)’ methods based on a single-determinantal reference
photoelectron spectroscopy has established the singlet–triplet (S-T) splitting to be −3.8 ±
0.5 kcal mol −1 (Wenthold et al. 1998). This corresponds to an energy splitting of −4.2 ±
0.5 kcal mol −1 (i.e., differences in zero-point vibrational energy have been removed). Highlevel
calculations suggest, not surprisingly, that the S-T splitting in the N-protonated pyridyne
system should be very nearly the same (Debbert and Cramer 2000). Table 7.3 illustrates the
results from a variety of different levels of electronic structure theory applied to computing
the S-T splitting using the cc-pVDZ basis set.
Notice, first, how spectacularly wrong the HF results are, this being indicative of significant
multireference character for the singlets, which should really be described as about 60:40
mixtures of the two determinants corresponding to double occupation of the antibonding and
bonding combinations of the σ orbitals. In the p-benzyne case, the MP2 calculation correctly
predicts the singlet to be the preferred state, but drastically overshoots in doing so. As is
typical, MP3 oscillates back to the HF prediction (triplet ground state) but with a smaller
margin of error, and then MP4 corrects back again in the proper direction, with a somewhat
smaller overestimation than was observed for MP2. Clearly, however, one could not with
confidence extrapolate to an infinite-order perturbation theory result from these four points.
The situation is much the same for the 2,5-pyridynium ion, except that the MP2 result is
very close to experiment. Such fortuitous agreement is obviously entirely coincidental, as
the perturbation series is wildly oscillating.
Note also the importance of triple excitations in correcting for the multideterminantal
character. The change in the S-T splitting from inclusion of the triples at the MP4 level
is as large as or larger than the change in going from MP3 to MP4SDQ. A similar effect
is seen with the QCISD and CCSD formalisms – both incorrectly predict triplet ground
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