Acetylene point group

The point group is derived from by the inclusion of U , therefore, all linear molecules with a plane of symmetry perpendicular to the axis belong to D f. Acetylene... 

Acetylene (HC=CH) belongs to the point group whose character table is given in Table A.37 in Appendix A, and its vibrations are illustrated in Figure 6.20. Since V3 is a vibration and T T ) = 2"+, the 3q transition is allowed and the transition moment is polarized along the z axis. Similarly, since Vj is a vibration, the 5q transition is allowed with the transition moment in the xy plane. 

The vibrations of acetylene provide an example of the so-called mutual exclusion mle. The mle states that, for a molecule with a centre of inversion, the fundamentals which are active in the Raman spectmm (g vibrations) are inactive in the infrared spectmm whereas those active in the infrared spectmm u vibrations) are inactive in the Raman spectmm that is, the two spectra are mutually exclusive. Flowever, there are some vibrations which are forbidden in both spectra, such as the torsional vibration of ethylene shown in Figure 6.23 in the >2 point group (Table A.32 in Appendix A) is the species of neither a translation nor a component of the polarizability. 

Let us first examine a few special cases that cover most common point groups. A linear molecule, such as HCN (point group Coov) or acetylene (Dxl), will lie along one principal axis, say the z axis, so that the first eigenvalue of the inertial tensor vanishes and the other is doubly degenerate alternatively, by the second case in Eq. 3 x, = v, = 0 for all i, and thus % = 0. 

Dcoh- The other point group of linear molecules, e.g., carbon dioxide and acetylene. 

Example 9.2-2 Acetylene HC=CH is a linear molecule with point group D,x,h and has 3N— 5=7 normal modes, which an analysis like that in Example 9.1-1 shows to be of symmetry... 

Symmetiy Relations. Each normal coordinate and every wavefunction involving products of the normal coordinates, must transform under the symmetry operations of the molecule as one of the symmetry species of the molecular point group. The ground-state function in Eq. (3 a) is a Gaussian exponential function that is quadratic in Q, and examination shows that this is of Xg symmetry for each normal coordinate, since it is unchanged by any of the symmetry operations. From group theory the symmetry of a product of two functions is deduced from the symmetry species for each function by a systematic procedure discussed in detail in Refs. 4, 5,7, and 9. The results for the D i, point group apphcable to acetylene can be summarized as follows ... 

FIGURE 6.5 Molecular orbital (MO) level diagram and the electron configuration for the cumulenic Cm carbon ring with Dsh point group symmetry (a) and the acetylenic carbon ring with Cgh point group symmetry (b). 

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