Degenerate vibrations operations

Geometric phase effect (GPE) (Continued) adiabatic states, conical intersections invariant operators, 735-737 Jahn-Teller theorem, 733-735 antilinear operator properties, 721-723 degenerate/near-degenerate vibration levels, 728-733... 

Thus if there are no degenerate vibrations, each normal coordinate is either unchanged or multiplied by — 1 upon application of a symmetry operation. Each Qk is a linear combination of the mass-weighted Cartesian displacement coordinates of the nuclei. If Qk is multiplied by — 1, each Cartesian displacement coordinate is multiplied by - 1, which reverses the directions of all the displacement vectors. If Qk is unchanged by a symmetry operation, then the symmetry operation sends the displacement vectors to a configuration indistinguishable from the original one. (The displacement vectors are defined relative to molecule-fixed axes, which in turn are defined relative to the nuclear positions. The effect of a symmetry... 

Objects which have C or S axes with n> 1 have degenerate vibrations, i.e., two (or more, for cubic or icosahedral point groups or linear molecules) vibrations have the same frequency. Carrying out a symmetry operation leads to a linear combination of degenerate vibrations. 

The correctness of transformation for the coordinates of degenerate vibrations differs from the A j coordinates because a covering operation applied to one of a pair of E coordinates does not give the same coordinate or its negative. Instead a linear combination of the two coordinates forming the pair is produced. For example, when the identity operation, /, is applied to... 

For a doubly degenerate normal mode, both components must be used together as the basis of a two-dimensional irreducible representation. For example, the operations C2 and ctv on the two normal vibrations that constitute the i>6 mode lead to the character (sum of the diagonal elements of the corresponding 2x2 matrix) of -2 and 0, respectively, as illustrated below. Working through the remaining symmetry operations, the symmetry species of can be identified as Eu. 

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