Essentials of Computational Chemistry

7.6 PRACTICAL ISSUES IN APPLICATION 231
this term is infinite! Note that this is not a case of the actual electronic state being degenerate,
but is entirely an artifact of using a single-reference wave function.
As a general rule, whenever the frontier orbital separation becomes small, the magnitude
of the MPn energy terms will become large because of their inverse dependence on orbital
energy separation, and perturbation theory will be very slowly convergent in such instances.
Such decreased separations also increase the degree of multireference character in the wave
function, so the tight coupling between these two phenomena is rationalized. Figure 7.5
provides an example of this phenomenon for the case of the energy of carbonyl oxide,
which has a moderate degree of multireference character, relative to its isomer dioxirane.
The comparatively small size of these systems permits the extension of perturbation theory
through fifth order with the 6-31G(d) basis set, but even the difference between the MP4
and MP5 results remains a fairly large 1.3 kcal mol −1 .
CCSD is similarly sensitive to multireference character, although it is less obvious that this
should be so based on the formalism presented above. However, inclusion of triples in the
CCSD wave function is usually very effective in correcting for a single-reference treatment
of a weakly to moderately multireference problem. Of course, the most common way to
include the triples is by perturbation theory, i.e., CCSD(T), and as noted above, this level
too can be unstable if singles amplitudes are large. In such an instance, BD(T) calculations,
which eliminate the singles amplitudes, can be efficacious.
To illustrate some of these points in greater detail, consider the aryne diradicals p-benzyne
and its isoelectronic but charged congener, N-protonated 2,5-pyridyne (Figure 7.6). In these
systems, the frontier orbitals of interest are the bonding and antibonding combinations of the
‘singly occupied’ σ orbitals left after abstraction of two hydrogen atoms from the aromatic
ring. Because the orbitals are para-related to one another, the interaction between them
is weak, and thus the frontier orbital energy separation is small. As such, the closed-shell
singlet and triplet states lie relatively near one another; in the case of p-benzyne, negative ion
Energy difference (kcal mol −1 )
−28
−30
−32
−34
−36
−38
−40
−42
0
1 2 3 4 5
Order of perturbation theory
6
H
O
O
H H
O O
Figure 7.5 Slowly oscillatory behavior of MPn/6-31G(d)//HF/6-31G(d) theory for the energy separation
between carbonyl oxide and dioxirane. Accurate extrapolation from this perturbation series is
an unlikely prospect
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H