Symmetry and degeneracy

In this paper, we review progress in the experimental detection and theoretical modeling of the normal modes of vibration of carbon nanotubes. Insofar as the theoretical calculations are concerned, a carbon nanotube is assumed to be an infinitely long cylinder with a mono-layer of hexagonally ordered carbon atoms in the tube wall. A carbon nanotube is, therefore, a one-dimensional system in which the cyclic boundary condition around the tube wall, as well as the periodic structure along the tube axis, determine the degeneracies and symmetry classes of the one-dimensional vibrational branches [1-3] and the electronic energy bands[4-12]. 

We then discover an extremely important fact each normal coordinate belongs to one of the irreducible representations of the point group of the molecule concerned and is a part of a basis which can be used to produce that representation. Because of their relationship with the normal coordinates, the vibrational wavefunctions associated with the fundamental vibrational energy levels also behave in the same way. We are therefore able to classify both the normal coordinates and fundamental vibrational wavefunctions according to their symmetry species and to predict from the character tables the degeneracies and symmetry types which can, in principle, exist. 

This result is tremendously useful, it not only leads to selection rules for vibrational spectroscopy but also, as was the case with electronic wavefunctions (see 8-2), allows us to predict from inspection of the character table the degeneracies and symmetries which are allowed for the fundamental vibrational wavefunctions of any particular molecule. 

As the strength of the interaction changes, states of the same spin degeneracy and symmetry cannot cross. 

P.-O. Lowdin and O. Goscinski Treatment of Constant of Motion, Degeneracies and Symmetry Properties hy Means of Multi-Dimensional Partitioning Int. J. Quantum Chem. 5, 685 (1971). 

Lowdin, P-O., 8cGoscinski, O. (1999). Studies in perturbation theory. XIV. Treatment of constants of motion, degeneracies and symmetry properties by means of multidimensional partitioning. International Journal of Quantum Chemistry, 5, 685. 

Studies in Perturbation Theory. XIV. Treatment of Constants of the Motion, Degeneracies and Symmetry Properties by Means of Multidimensional Partitioning... 

The coupling of vibrational and electronic motions in degenerate electronic states in inorganic complexes, part II states of triple degeneracy and systems of lower symmetry. A. D. Liehr, Prog. Inorg. 

Suppose now that A) and B) belong to an electronic representation I ,. Since H is totally symmetric, Eq. (6) implies that the matrix elements (A II TB) belong to the representation of symmetrized or anti-symmetrized products of the bras (A with the kets 7 A). However, the set TA) is, however, simply a reordering of the set ( A). Hence, the symmetry of the matrix elements in the even- and odd-electron cases is given, respectively, by the symmetrized [Ye x Te] and antisymmetrized Ff x I parts of the direct product of I , with itself. A final consideration is that coordinates belonging to the totally symmetric representation, To, cannot break any symmetry determined degeneracy. The symmetries of the Jahn-Teller active modes are therefore given by... 

Boriskina, S.V., 2005, Symmetry, degeneracy and optical confinement of modes in coupled microdisk resonators and photonic crystal cavities, submitted to J. Quantum Electron. 

Tetragonal distortion from octahedral symmetry often occurs even when ail six ligands of a complex are the same Two L groups that are irans to each other are found to be either closer to or farther from ihe metal ion than are the other four ligands. A distortion of this type actually is favored by certain conditions described by the Jahn-Teller theorem. The theorem stales that for a nonlinear molecule in an electronically degenerate state, distortion must occur to lower the symmetry, remove the degeneracy, and lower the energy.47 We can determine which octahedral complexes... 

Attention should be paid to degeneracies and to the symmetry and equivalence properties of the MOs. 

If the symmetry of a molecule is lowered by a perturbation (for example by the substitution of a foreign ion in a crystal lattice) this may remove degeneracies and/or permit transitions that were forbidden in the more symmetric molecule. 

A study of the spin-allowed bands in this complex has revealed that there is no observable phase-coupling effect [50]. According to Atanasov et al. [50] this must be explained by the near-degeneracy of donor levels of tft and % symmetry type. We consider it also conceivable that coordination of the metal ion polarizes the charge distribution in the ligand chain as depicted below ... 

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