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Theoretical Background

Motivation: Why measure the degree of symmetry?

The concept of symmetry has attracted virtually all domains of intellectual activity, and has strongly influenced the sciences and the arts. The concept has functioned as a condensed language for the description and classification of order within shapes and structures; as an identifier of inherent correlations between structure and physical properties of matter; and as a guideline in artistic and practical aesthetic design.

However, in reality, exact symmetry is a rare realization of Nature, beyond the atomic level. To appreciate this fact one should refine the resolution of observation - spatial or temporal - up to the point where it becomes evident. The advent of highly sensitive analytical and probing tools in modern chemistry shows again and again that even structures which have classically been treated as symmetric, actually are not.

Consider, for instance, the observation of symmetric molecules on time scales that are faster than typical vibration rates. Even molecules such as CH4 will "never" appear as perfectly tetrahedral (Fig. 1). Or consider what happens to symmetric molecules in condensed phases: absorb a perfectly symmetric molecule on a surface, and its original symmetry is removed (Fig. 2).

Figure 1

Figure 2

Next consider the rotation of the two tetrahedra of ethane, H3C-CH3, around the C-C bond (Fig. 3). Current wisdom allows an exceedingly poor description of that process from the symmetry point of view: Ethane is D3d when staggered, D3h when eclipsed and D3 anywhere in between. But let us observe the rotamer which is only 5 degrees away from any of the extremes. Is it already a D3 species? (Fig. 3).

Figure 3

In another example, consider the case of two ethylenes approaching each other for a [2+2] reaction (Fig. 4). The answer to the question of whether that reaction is allowed thermally or photochemically, or whether a suprafacial or antarafacial process will take place, or whether the reaction will take place at all, is very much dependent on the symmetry of alignment of the two reacting molecules or double bond moieties. The ideal symmetry needed for a suprafacial photochemical formation of cyclobutane is the parallel approach of the two double bonds in a D2h symmetry. In many cases, however, the two ethylenes are not in an ideal D2h alignment because of an intramolecular frozen conformation of the two double-bonds, or because of non-symmetric steric hindrance caused by substituents on the double bonds; yet current wisdom is to treat these case as if they were in the needed ideal symmetry, ignoring the fact that in these common situations, only a vague memory of the ideal symmetry exists.

Figure 4

Finally consider the case of hexachlorobenzene vs. bromo-pentachlorobenzene (Fig. 5): The first has a D6h symmetry, but the latter jumps to C2v. Isn't it more natural to treat the symmetry of bromo-pentachlorobenzene as resembling, to some degree, a D6h object?

Figure 5

It therefore seems natural to evaluate how much of a given symmetry there is in a structure, or, in other words, to treat symmetry as a structural property of continuous behavior. Such a continuous symmetry scale should be able to express quantitatively the degree of content of a given symmetry in any (symmetry-distorted) structure. A general symmetry measurement tool has been designed towards this goal.

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