Profiting from symmetry

Large parts of the search tree can be pruned in cases of higher symmetry. If two labelings result in the same bond matrix at different positions in the tree, then a symmetry (automorphism) has been found. The information on automorphisms that accumulates in the process finally defines the complete automorphism group of the graph or molecule. This is stored in the form of a set of generators (a Sims chain [93,142,296]). This information is used to prune parts of the backtrack tree found to be equivalent to other parts already considered. 

By backtracking, marking atom 5 on btD and refining by 5, candidate 2 is found and the renumbering scheme is now  

Backtracking and mru-king atom 5 on btl2 results in candidate 4, again automorphic to candidate 1, and as before a branch is now pruned. 

Note that only the renumbering schemes, not the matrices are stored, as evident from the above examples. This results in a rather low memory requirement. A detailed description of the canonization procedure was given in (17,18]. 

At present MOLG EN-CiD only treats covalently bonded compounds i.e. made either of one or several components (connected or disconnected undirected graphs). Stereoiso- 

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Table 5.6. Profiting from symmetry for cubane, structure 4, Figure 5.8. Atoms that become unique are printed in bold.

Finally, vre should note that the LCAO-MO approximations profit from the high symmetry of some species and that application to less symmetrical... 

In view of the magnitude of crystal-field effects it is not surprising that the spectra of actinide ions are sensitive to the latter s environment and, in contrast to the lanthanides, may change drastically from one compound to another. Unfortunately, because of the complexity of the spectra and the low symmetry of many of the complexes, spectra are not easily used as a means of deducing stereochemistry except when used as fingerprints for comparison with spectra of previously characterized compounds. However, the dependence on ligand concentration of the positions and intensities, especially of the charge-transfer bands, can profitably be used to estimate stability constants. 

One might suppose it would be easier, and perhaps more profitable, to examine, observe, and experiment, than to turn one s eyes inwards with the hope of discovering exactly "the road followed by the Great Architect of the Universe in the creation of the world." But the alchemical method found it easier to begin by introspection. The alchemist spun his universe from his own ideas of order, symmetry, and simplicity, as the spider spins her web from her own substance. 

In a considerable number of cases both sets of modes have been observed in on-specular VEEL spectra, and the deduction has been made that the symmetry of the surface complex is Cs (or less) (145,146,151,152,160,162). The question remains whether this implies a twofold bridged adsorption site or a neighbor-induced asymmetry within an essentially C3 site, as already described. However, there are examples of species on Pt(lll) (150), Ni(ll 1) (117), and Cu(lll) (161) surfaces for which MSSR as applied to VEEL spectra clearly indicates C3v symmetry of the surface complex, without significant differences in the other frequencies as observed off-specular. These favorable cases may arise from particularly regular arrays of adsorbed species, the presence of which could very profitably be confirmed by LEED. We deduce that the CH3 adsorption sites are intrinsically C3v as far as the bare surface is concerned, i.e., on-top or threefold hollow in nature with the threefold axis of the CH3 group perpendicular to the surface. 

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