Vibration degenerate

In a molecule belonging to a degenerate point group, for example C, , the non-degenerate vibrations of the various sets of equivalent nuclei can be treated as in Section 6.2.2.1. 

These nuclei lie on the planes of symmetry and there are three nuclei belonging to each set. The three nuclei have nine degrees of freedom of which two belong to the uj species and one to the 2 species. The nuclei have six degenerate degrees of freedom 

Species Degrees of freedom Degrees of freedom Number of normal vibrations 

In Table 6.6 the results for the point group are summarized and the translational and rotational degrees of freedom are subtracted to give, in the final column, the number of vibrations of each symmetry species. 

Species Degrees of freedom3 Degrees of freedom11 Number of normal vibrations 

In order to illustrate the vibrational motions of a molecule belonging to a non-commutative symmetry point group, we return to the considerations of Section 2.3.2 and once more use as our example the square-planar complex, NiFj. A non-linear penta-atomic molecule has nine independent vibrational coordinates, distributed among the symmetry species of T 4h. These can be fully specified by standard methods [7], but the following simple qualitative considerations allow us to conclude that there are seven in-plane and two out-of-plane vibrations. Fig. 4.10 depicts several of the in-plane modes the motion of the nickel atom to conserve the center of mass is implied. 

The sum of the four internal angles is 360°, so there can be no totally symmetric in-plane bending mode, in which all four increase together. The one non-degenerate in-plane bending mode is the antisymmetric bend (625) which 

The two out-plane-modes (not illustrated in the figure, are also easy to characterize they are both non-degenerate In one a2u)-, the F atoms move together parallel to 2r, and the central Ni atom moves in the opposite direction so as to conserve the center of mass. In the other (62 ), one pair of rans-situated atoms moves up, the other pair moves down, and the central atom stays put. 

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If the states are degenerate rather than of different symmetry, the model Hamiltonian becomes the Jahn-Teller model Hamiltonian. For example, in many point groups D and so a doubly degenerate electronic state can interact with a doubly degenerate vibrational mode. In this, the x e Jahn-Teller effect the first-order Hamiltonian is then [65]... 

Appendix D Degenerate and Near-Degenerate Vibrational Levels Appendix E Adiabatic States in the Vicinity of a Conical Intersection 1, Jahn-Teller Theorem... 

APPENDIX D DEGENERATE AND NEAR-DEGENERATE VIBRATIONAL LEVELS... 

Now, consider the general case of a V2 multiply excited degenerate vibrational level where V2 > 2, which is dealt with by solving the Schrddinger equation for the isotropic 2D harmonic oscillator with the Hamiltonian assuming the fonn [95]... 

Tables for all degenerate point groups, giving the symmetry species of vibrational combination states resulting from the excitation of one quantum of each of two different degenerate vibrations and of vibrational overtone states resulting from the excitation of two quanta of the same degenerate vibration, are given in the books by Herzberg and by Hollas, referred to in the bibliography.

For an anharmonic oscillator with degenerate vibrations the term values are modified from those of Equation (6.88) to... 

Lynden-Bell R. M. The effect of molecular reorientation on the line-shapes of degenerate vibrations in infra-red and Raman spectra of liquids, Mol. Phys. 31, 1653-62 (1976). 

This function decreases monotonically with increasing vibrational quantum number n, and hence an inverted vibrational distribution can never be described with a temperature (except for degenerate vibrations). A P n) distribution that is thermal , or at least not inverted, is indicative of a well on the PES that is connected with little or no barrier to the product asymptote. 

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... 

Irreducible representations (IRREPs), permutational symmetry degenerate/near-degenerate vibrational levels, 728-733... 

Longuet-Higgins phase-based treatment, two-dimensional two-surface system, scattering calculation, 154-155 three-state molecular system, 134-137 two-state molecular system, single conical intersection solution, 98-101 permutational symmetry, degenerate/near-degenerate vibrational levels, 730-733 Polyene molecules ... 

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