Essentials of Computational Chemistry

234 7 INCLUDING ELECTRON CORRELATION IN MO THEORY
Insofar as CASPT2 uses a multiconfigurational reference, one might expect it to be less
prone than MP2 to instability. This is entirely true, so long as the MCSCF reference is
adequate. If the MCSCF has converged to a spurious solution, perturbation theory is often
successful in identifying this because a very large contribution from one or more excitations
will be observed. Alternatively, if the MCSCF failed to include one or more critical orbitals,
again, large contributions will be obtained for corresponding excitations. From a formal
standpoint, it is better to include those orbitals in the active space than it is to rely on
CASPT2 to correct for both dynamical and non-dynamical behavior, even though in some
instances it seems the latter approach can give good results.
An alternative way to account for multiconfigurational character in the singlet is to
generate it using SF-CIS(D) from the triplet reference. Slipchenko and Krylov (2002) have
done this for p-benzyne using the 6-31G(d) basis set and computed a S−T splitting of
−2.2 kcal mol −1 . The SF-CIS(D) model is thus nearly as accurate as those including estimates
for triple excitations even though it is substantially less computationally expensive.
A separate issue that can contribute to instability in correlated calculations is spin contamination.
As noted in Chapter 6, spin contamination refers to the inclusion in the wave function
of contributions from states of higher spin that mix in when unrestricted methods permit
α and β spin orbitals to localize in different regions of space. As a rough rule, the sensitivity
of different methods to spin contamination is about what it is to multiconfigurational
character: MPn methods are to be avoided and inclusion of triples in CCSD or QCISD (or
BD) is important. So-called projected MPn methods attempt to correct for spin contamination
after the fact by projecting out states of higher spin from the correlated wave function
(see Appendix B), and these methods tend to be helpful in cases where spin contamination
is relatively small, say no more than 10% (Chen and Schlegel 1994). Unfortunately,
analytic gradients are not available for spin-projected methods, so they must be applied to
geometries the optimization of which may have taken place at a considerably less reliable
level.
An issue related to spin contamination is so-called Hartree–Fock instability. Various wave
functions can exhibit different kinds of instabilities, often, but not always, associated with
trying to describe a multiconfigurational system with a single-determinant approach. Thus, for
instance, RHF solutions may be unstable with respect to breaking the identical character of
the α and β orbitals – a so-called RHF → UHF instability (the UHF singlet wave function is
usually highly contaminated with triplet character after reoptimization). UHF wave functions
for symmetric systems can also be unstable, in this case with respect to spatial symmetry
breaking of the individual orbitals. The MOs, if allowed to relax, fail to fall into the irreps
of the molecular point group, adopting instead lower symmetry shapes even if the molecular
framework is held fixed so as to continue to belong to the higher symmetry point group.
Such instability tends to be associated with systems having delocalized spin – the allyl
radical is a classical example. All of these cases prove very problematic for perturbation
theory, but are handled with somewhat greater success by other correlated methods. In
certain very highly symmetric systems, the wave function can also be unstable to using
complex instead of real MOs, but this situation is rare. Most modern electronic-structure
programs allow one to check the stability of the HF wave function with respect to these