Q.181. Consider the possible vibration modes in the following linear molecules: 
(a) CO₂ (O — C—O); (b) C₂H₂ (H—C —C—H). 

(a) CO₂ (O - C - O)
The molecule has 9 degrees of freedom 3 for each atom. This means that it can have up to nine frequencies. 3 degrees of freedom correspond to rigid translation, the frequency associated with this is zero as the potential energy of the system can not change under rigid translation. The P.E. will not change under rotations about axes passing through the C-atom and perpendicular to the O - C - O line. Thus there can be at most four non-zero frequencies. We must look for modes different from the above.
One mode is 
Another mode is 
These are the only collinear modes.
A third mode is doubly degenerate: 
(vibration in and $\perp$  to the plane of paper).
(b) C₂H₂ ( H - C - C - H )
There are 4 x 3 - 3 - 2 = 7 different vibrations. There are three collinear modes.



Two other doubly degenerate frequencies are


together with their counterparts in the plane $\perp$ to the paper.