Symmetry theorems

Mezey, P.G. (1991) New symmetry theorems and similarity rules for transition structures. In Theoretical and Computational Models for Organic Chemistry, Formosinho, SJ. Csizmadia, I.G. and Amaut, L.G. (Eds.), Kluwer Academic Publishers, Dordrecht. 

In the present case, the extension of the scalar counting rules for n(S) to symmetry theorems is straightforwardly achieved by replacing n(S), n(v), n(e) n( > ) by the permutation representations F,T(S), r,T(v), r,T(e) and ra(vj) [13]. A permutation representation /j/v), of a set of objects x has character y(R) under operation (R) of the symmetry group of the undistorted framework, where x(R) is equal to the number of objects wnshifted under operation R. The subscript a is often dropped if there is no danger of confusion. With these replacements, equation (4) becomes... 

Mezey, P.G. (1990a). A Global Approach to Molecular Symmetry Theorems on Symmetry Relations Between Ground- and Excited-State Configurations. J.Am.Chem.Soc., 112, 3791-3802. 

IV. Point Group Symmetry Theorems Within the Tensor Surface Harmonic... 

Application of the Symmetry Theorems to the Skeletal Molecular Orbitals of... 

Trivalent Polyhedra The dual of a deltahedron is a trivalent polyhedron, meaning that every vertex is connected to three nearest neighbours. The fullerene networks of carbon are usually trivalent polyhedra. This reflects the sp hybridization of carbon, which can form three specialized forms of the Euler symmetry theorem can be formulated. We may start from Eq. (6.138) and replace vertices by faces. The edge terms remain the same since they are totally symmetric under the local symmetries of the edges. Rotations of the edges by 90° will thus not affect these terms. 

Fig. 6.9 Edge bonding in electron-precise trivalent cages. The valence shell splits into an occupied set of localized edge-bonds, and a matching virtual set of edge-anti-bonds. The sets may be further differentiated by use of the symmetry theorems...

NEW SYMMETRY THEOREMS AND SIMH ARITY RULES FOR TRANSITION STRUCTURES... 

In this chapter some implications of a family of new symmetry theorems will be described, with particular emphasis on transition structure problems. The results have both theoretical and computational consequences for conformational analysis, shape analysis, and reaction path studies in organic chemistry. We shall use the concepts, terminology and notations reviewed in earlier works [1-3]. The reader may find... 

Symmetry Constraints on Transition Structures from Global Symmetry Theorems... 

In order to prove this result, we shall show that the catchment regions C(X,i) and C (X,, i ) overlap, and we shall apply the catchment region point symmetry theorem to C (X, i ) and to a point K from the overlapping region. 

The first of these theorems, the catchment region point symmetry theorem states the following ... 

The important observation is that the theorem does not require the choice of this maximal common point group of B and one doesn t have to include all common symmetry elements in family R the vertical point symmetry theorem of nuclear configuration spaces is valid even if one includes only some of the eligible symmetry elements in families R and R. ... 

Mezey PG (1990) A global approach to molecular symmetry theorems on symmetry relations between ground and exeited state configurations. J Am Chem Soc 112 3791-3802... 

R-SLF and R-PDLF techniques. Levitt and co-workers have presented symmetry theorems that are useful in the design of rf schemes for selective recoupling and decoupling in MAS solid-state NMR [36]. For example, it was found that certain R-type sequences [60] lead to reactivation of heteronuclear dipolar couplings... 

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