Topological symmetry

Among new concepts that have been put forward are the relationships between symmetry, topology, and aromaticity, and bond stretch isomerism and polytopal rearrangements. ... 

For states of different symmetry, to first order the terms AW and W[2 are independent. When they both go to zero, there is a conical intersection. To connect this to Section III.C, take Qq to be at the conical intersection. The gradient difference vector in Eq. f75) is then a linear combination of the symmetric modes, while the non-adiabatic coupling vector inEq. (76) is a linear combination of the appropriate nonsymmetric modes. States of the same symmetry may also foiiti a conical intersection. In this case it is, however, not possible to say a priori which modes are responsible for the coupling. All totally symmetric modes may couple on- or off-diagonal, and the magnitudes of the coupling determine the topology. 

Molecular Orbitals Possess Specific Topology, Symmetry, and Energy-Level Patterns... 

In this chapter the symmetry properties of atomie, hybrid, and moleeular orbitals are treated. It is important to keep in mind that both symmetry and eharaeteristies of orbital energetics and bonding "topology", as embodied in the orbital energies themselyes and the interaetions (i.e., hj yalues) among the orbitals, are inyolyed in determining the pattern of moleeular orbitals that arise in a partieular moleeule. 

The perturbation of the PMD symmetry is accompanied by a decrease in the charge alternation and by the appearance of bond alternation from one end group to another. The bond alternation ampHtude has been revealed to be proportional to the asymmetry degree, which can be calculated as the difference of topological indexes = 4>gj — 4>gg. The effect is maximum if Tgj > 45° and Tgg < 45°. If A4>j2 = 90°, the ideal polyene state is... 

In a general concept of a symmetry-restricted anharmonic theory Krumhansl relates the phonon anomalies to the electron band topology. The latter is directly determined by the competition of nearest neighbour interactions which in turn can be a function of stress, composition and temperature Nagasawa, Yoshida Makita simulated the <110> ... 

A few class-2 —> class-3 transitions are Cr-symmetry dependent. For example, while the observed behavior of certain rules on 3 may be different from 2, the dynamics on all intermediate and succeeding topologies may, in fact, remain of the... 

KinR81 King, R. B. Chemical applications of topology and group theory. 9. The symmetries of coordination polyhedra. Inorg. Chem. 20 (1981) 363-372. 

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