Symmetry Breaking

C. David Sherrill
School of Chemistry and Biochemistry
Georgia Institute of Technology
Atlanta, GA 30032-0400

Introduction

In three publications, we have presented some interesting new results for ``symmetry breaking'' molecules. In the context of molecular electronic structure theory, the term symmetry breaking usually implies that a molecule turns out to have a lower point-group symmetry than expected. In particular, degenerate electronic states may become more stabilized by a geometrical distortion that breaks the electronic degeneracy (the Jahn-Teller effect). Unfortunately, however, theoretical electronic structure methods seem prone to predicting such symmetry lowerings even when they do not occur in reality. We term this artifactual symmetry breaking.

Thus an obvious question is how to determine whether theoretically predicted symmetry breaking is real or artifactual. The first thing one should examine is the symmetry of the wavefunction at the high-symmetry geometry: if the wavefunction fails to exhibit the full symmetry of the point group, this is a sign that predictions of geometrical symmetry breaking may be artifactual. In such cases, one might say that the wavefunction itself exhibits artifactual symmetry breaking. Paradoxically, for a given approximation scheme, such symmetry broken wavefunctions can be lower in energy (and hence better in a variational sense) than symmetric wavefunctions, even though the exact wavefunction must be symmetric.

This symmetry breaking problem is actually rather widespread and can easily make a project become completely intractable. Radicals are particularly prone to symmetry breaking and related problems. Symmetry breaking is frequently the cause of completely unphysical vibrational frequency predictions (e.g., frequencies of 10,000 cm^-1 or more). Many users of quantum chemistry programs probably assume that all such results are due to bugs in the code (or failing to follow the correct solution for one of the geometries, which frequently happens), but this is not always true: they may result from fundamental failures in the underlying quantum mechanical models.

Symmetry Breaking and the OO-CCD Method

We found that the atmospherically important O₄⁺ molecule exhibits unrealistically large vibrational frequencies at the coupled-cluster singles and doubles (CCSD) level of theory. This is a very interesting result, since Crawford et al. have indicated that CCSD is highly resistant to these problems, even more so than the more complete CCSD(T) method [J. Chem. Phys. 107, 10626 (1997)]. Using our new implementation of the optimized-orbital coupled-cluster doubles (OO-CCD) method [J. Chem. Phys. 109 4171 (1998); (Abstract)], we obtain very reasonable results. This is in accord with previous work by Stanton et al. and by Barnes and Lindh which indicates that Brueckner CCD (to which our OO-CCD is numerically very similar) sometimes cleans up certain types of symmetry breaking problems.

6-31G* Geometries and Frequencies for Quartet O₄⁺

Method	r(O-O)	r(CM)	a_g	a_g	b_1g	a_u	b_2u	b_3u
UHF	1.1176	2.4058	2242	255	396	197	-543	3223
ROHF	1.1152	2.4073	2256	258	417	199	83	3612
UHF CCSD	1.1737	2.3792	1808	269	341	178	154	1726
ROHF CCSD	1.1730	2.3793	1813	269	372	179	-66	2036
B-CCD	1.1723	2.3794	1821	269	342	179	82	1194
OO-CCD	1.1728	2.3793	1816	269	372	179	84	1193
Expt.								1320

Symmetry Breaking and Density Functional Theory

In another paper [Chem. Phys. Lett. 302, 425 (1999); (Abstract)], we examined many different versions of Kohn-Sham density functional theory (DFT) as applied to three molecules which exhibit artifactual symmetry breaking in the unrestricted Hatree-Fock (UHF) wavefunction: O₂⁺, O₄⁺, and NO₃. We found that all DFT methods considered yield symmetric densities for each of the three molecules unless we use hybrid DFT functionals mixing in an unusually large fraction of pure Hartree-Fock exchange. Further tests indicated that exchange is more important than correlation in determining whether symmetry is preserved or broken.

The above results were very encouraging and suggested that perhaps DFT may be a reliable method to use for symmetry breaking molecules. However, to test this hypothesis, it is necessary to assess the reliability of DFT for molecular properties (such as geometries and vibrational frequencies) of molecules prone to artifactual symmetry breaking. We recently published the results of such a study [J. Chem. Phys. 114, 8257 (2001); (Abstract)]. We found that DFT appears to be among the more robust methods available for such challenging cases, but it is still not immune to anomalous predictions for molecular properties.

This material is based in part upon work supported by the National Science Foundation under Grant No. 0094088. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.