Permutation implications

Heinemann, U., and Hahn, M. (1995). Circular permutation of polypeptide chains implications for protein folding and stability. Prog. Biophys. Mol. Biol, 64, 121-143. 

The ultimate implication should be clear. A given set of sub-atomic particles may be conditioned in an infinite variety of ways by different environments and histories. All of these constraints should be stipulated in the molecular Hamiltonian in order to avoid confusion with irrelevant permutations. The degree of structure becomes a function of the complete Hamiltonian. 

Pasteur clearly saw the molecular implications of his discovery. But what feature, at the molecular level, mimicked the asymmetry of the tartrate crystals Le Bel and van t Hoff suggested that, because the valency of a carbon atom is 4, it could be represented as occupying the center of a tetrahedron, at the apices of which are other atoms. Where the atoms at the apices are all different, the result is an asymmetric arrangement of which there can be a number of permutations. Different arrangements... 

Dynamic processes causing ligand permutation in pentacoordinate phosphorus compounds constitute a well-studied area (148). As illustrated in Sect. V-A, various authenticated penta- or hexacoordinate organosilanes have been reported. However studies of the isomerization processes of these compounds are limited. Such processes are of importance, since penta- or hexacoordinate intermediates are implicated in substitution reactions or in racemization of tetra-coordinate silicon compounds. The stereochemical outcome of such reactions is therefore related to the configurational stability of penta- or hexacoordinate intermediates. 

The interpretation of Eq. (1.2) in terms of permutational symmetry of determinants is clear. To make the analogous content of Eqs. (1.4) and (1.5) more transparent, we now examine some of the implications that follow from these equations. Let us first examine Eq. (1.5). For r = s, this reads rr+ — 1. When operating on a ket in which is occupied, the first... 

Lijnen, E., Ceulemans, A. The permutational symmetry of the icosahedral orbital quintuplet and its implication for vibronic interactions. Europhys. Lett. 80,67006 (2007)... 

In discussing the implications of these results, it is helpful to be able to refer to the various permutational isomers by the sorts of designations now commonly applied to pentacovalent phosphorus species (37,38). In equation 9 we show reaction of a cis isomer, 12, (generalized to represent a ring of any size) to give the permutamer, 13. This isomer, in the formalism employed in discussion of pentacovalent phosphorus intermediates, results... 

Collins, M. A., Parsons, D. F. (1993). Implications of rotation-inversion-permutation invariance for analytic molecular potential energy surfaces. The Journal of Chemical Physics, 99, 6756. 

The potential of such chemical shift calculations may be further illustrated by a recent application to nido-C4B Hu (12, see Figure 4). Three isomers of this carbaborane are known which have the same basic stmcture of a decapitated icosahedron, but which differ in the relative positions of the carbon atoms on that cage. Many more isomers (i.e., permutations of four carbons on 11 sites) are possible, and in fact, the computations have revealed that only one of the previously proposed stmctures had been correct. Two structural assignments have to be revised, e.g., 12b instead of 12a (Figure 4). This has important implications regarding... 

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