Broken symmetry analysis

Noodleman, L., Case, D.A., and Aizman, A. (1988) Broken symmetry analysis of spin coupling in iron-sulfur clusters. Journal of the American Chemical Society, 110, 1001-1005. 

Broken Symmetry Analysis for a Three-Iron Cluster. 

Cheletropic processes are defined as reactions in which two bonds are broken at a single atom. Concerted cheletropic reactions are subject to orbital symmetry analysis in the same way as cycloadditions and sigmatropic processes. In the elimination processes of interest here, the atom X is normally bound to other atoms in such a way that elimination gives rise to a stable molecule. In particular, elimination of S02, N2, or CO from five-membered 3,4-unsaturated rings can be a facile process. 

It will be argued that any interaction represents a broken symmetry and that mathematically precise symmetry never occurs in the real world. The implication is that the almost magical parallel between mathematics and physics that fascinates both scientists and philosophers, breaks down at the most fundamental level. This observation has little or no impact on the analysis of physical systems (e.g. in crystallography) where exact symmetry is no more than a simplifying model. However, the common dogma of time-reversible laws of physics, may in the view of all this, introduce a fallacy that re-emerges as the paradox of irreversability only much later. 

We also performed symmetry-adapted calculations for the hexuply bonded configuration. While the calculated values of and are reasonably close to the broken-symmetry results, the vibrational frequency is far too high and the anharmonicity far too low. In the final analysis, Cr2 should be regarded neither as a hexuply bonded nor as an antiferromagnetic dimer. Its correct description... 

The question posed in the preceding paragraphs as to the need for a reevaluation of the concept of allowedness , can be dismissed as a non-problem as long as it is taken as axiomatic that, for an orbital symmetry analysis to be of any use, the symmetry elements [retained along the pathway] must bisect bonds made or broken in the process . In contrast to the allowed conrotatory cyclization of butadiene to cyclobutene, in which the C2 axis bisects a newly formed cr bond, the only bond bisected by the axis in its conversion to bicyclobutane is the one between C2 and C3, which is essentially single in both the reactant and the product. 

Mapping Analysis Based on Broken-Symmetry States... 

In the case of a symmetry-broken solution, these weights can not only diagnose the problem, but also quantify its extent. As for the projection operator considered in Section 3.2, this analysis is used primarily with Abelian point groups. 

---