Artifactual symmetry breaking

A curious effect, prone to appear in near degeneracy situations, is the artifactual symmetry breaking of the electronic wave function [27]. This effect happens when the electronic wave function is unable to reflect the nuclear framework symmetry of the molecule. In principle, an approximate electronic wave function will break symmetry due to the lack of some kind of non-dynamical correlation. A typical example of this case is the allyl radical, which has C2v point group symmetry. If one removes the spatial and spin constraints of its ROHF wave function, a lower energy symmetry broken (Cs) solution is obtained. However, if one performs a simple CASSCF or a SCVB [28] calculation in the valence pi space, the symmetry breaking disappears. On the other hand, from the classical VB point of view, the bonding of the allyl radical is represented as a superposition of two resonant structures. 

Borden considers his group s work on the importance of the inclusion of dynamic electron correlation in many different types of problems to be his most significant contribution to computational methodology. In addition, his work on the Cope rearrangement and artifactual symmetry breaking taught him to question the applicability of routine methods to what first appears to be a routine problem but one that later turns out to require high level calculations. 

Perhaps the greatest need for Brueckner-orbital-based methods arises in systems suffering from artifactual symmetry-breaking orbital instabili-ties, " ° where the approximate wavefunction fails to maintain the selected spin and/or spatial symmetry characteristics of the exact wavefunction. Such instabilities arise in SCF-like wavefunctions as a result of a competition between valence-bond-like solutions to the Hartree-Fock equations these solutions typically allow for localization of an unpaired electron onto one of two or more symmetry-equivalent atoms in the molecule. In the ground Ilg state of O2, for example, a pair of symmetry-broken Hartree-Fock wavefunctions may be constructed with the unpaired electron localized onto one oxygen atom or the other. Though symmetry-broken wavefunctions have sometimes been exploited to produce providentially correct results in a few systems, they are often not beneficial or even acceptable, and the question of whether to relax constraints in the presence of an instability was originally described by Lowdin as the symmetry dilemma. ... 

If it were possible to carry out full Cl calculations (cf. Eq. [4] and the paragraph that follows it), artifactual symmetry breaking would, of course, not occur. Moreover, the variational principle would ensure that pure spin states were obtained, irrespective of whether one started with ROHF or UHF MOs, because in a full Cl calculation, the nature of the starting MOs is irrelevant. Although full Cl calculations are rarely practical, perhaps one might hope to reap some of their benefits without paying the entire cost. A method based on this idea is the so-called CASSCF (complete active space SCF) procedure. ... 

This example shows why performing calculations with wave functions that do not show artifactual symmetry breaking is of critical importance. When CASSCF or n Cl wave functions for TMM are used, so that artifactual symmetry... 

A reliable electronic-structure model which provides a balanced description of the electronic states of interest is required for the computation of inter-state vibronic coupling constants. In particular, artifactual symmetry-breaking effects in the... 

Another category of approaches that avoids the symmetry breaking problem of the orbitals for the target state is based on using a related, well-behaved HF reference instead. Examples of such techniques include quasi-restricted Hartree-Fock coupled-cluster (QRHF CC) (11,19), symmetry adapted cluster configuration interaction (SAC-CI) (22,23), coupled-cluster linear response theory (CCLRT) (24-26) or equivalently equation-of-motion coupled-cluster (EOM-CC) (27-32), Fock space multi-reference coupled-cluster (FSMRCC) (33-37), and similarity transformed equation-of-motion coupled-cluster (STEOM-CC) (38-40). In the case of NO3 and N03, the restricted Hartree-Fock (RHF) orbitals of the closed-shell N03 anion system can be used as the reference. The anion orbitals are stable with respect to symmetry perturbations, and the system does not suffer from the artifactual symmetry breaking of the neutral and cation. 

The NO3 vertical ionization spectrum was calculated by the DIP-STEOM-CCSD and DIP-EOM-CCSD methods. These methods avoid artifactual symmetry breaking of the reference wavefunction by starting from the symmetry-correct nitrate anion orbitals and provide a balanced treatment of dynamical and non-dynamical correlation effects. In general, the DIP-STEOM and DIP-EOM results agree well with the experiment of Wang (5) and with previous theoretical assignments (5,7). However, in our calculations, the vertical transition to the E" state nearly coincides with that of the E" state, in contrast to previous studies. Our calculations therefore do not support the assignment of the broad experimental feature near 14.05 eV to the E" state. 

However, as we will show below, a molecule possessing C2v symmetry cannot possess two degenerate resonance hybrids. Thus, the allyl radical has a three electron pi bond, which cannot be described properly at the MO monoconfigurational level. This type of symmetry breaking is called artifactual . 

E. R. Davidson and W. T. Borden, /. Phys. Chem., 87, 4783 (1983). Symmetry Breaking in Polyatomic Molecules Real and Artifactual. See also T. Bally and W. T. Borden, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., Wiley-VCH, New York, Vol. 13, pp. 1-97. Calculations on Open-Shell Molecules A Beginner s Guide. 

Finally, we should mention that the localization of spin (and charge in the case of radical ions), which occurs as a consequence of symmetry breaking, does not require the presence of molecular symmetry. The two valence structures that arise by localization of spin in different parts of a molecule need not necessarily be identical, as they are in allyl radical. Since ROHF has a general tendency to localize unpaired electrons in open-shell molecules, where higher level calculations show this localization to be artifactual, one should, in general, avoid using ROHF to optimize geometries. 

E. R. Davidson and W. T. Borden,/. Phys. Chem., 87, 4783 (1983). Symmetry Breaking in Polyatomic Molecules Real and Artifactual. 

Tor example, molecules with higher than twofold symmetry axes may have degenerate states whose wavefunctions necessarily break symmetry, as a consequence of the Jahn-Teller theorem. In these cases singularities (so-called conical intersections) arise as a result of state crossings and are not artifactual. State crossings can also occur accidentally, when only a plane or a twofold axis of symmetry is present, and the Jahn-Teller-type effects that result also create conical intersections on potential energy surfaces. 

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