Vibrational Modes or Degrees of Freedom:

A diatomic molecule has only one possible way of executing vibrational motion because there is only one bond and the molecule is linear.  A molecule containing more than two atoms has more than one bond and also these bonds may be at angle with respect to each other. For example, water is a triatomic molecule having two O -     H bonds at an angle of about 104.5^o. Such a molecule can vibrate in more than one possible way, like both the bonds of the molecule may vibrate symmetrically i.e., in unison or they may be showing an asymmetric vibration or the vibration may involve a change in the bond angle. How do we ascertain the possible number of such modes of vibration for a larger molecule? The discussion that follows will help to give an answer.

For a molecule containing N atoms we need to specify 3N coordinates (three for each atom) so as to completely describe the molecule. This fixes all the bond lengths and angles for the molecule.  We may say that the molecule has 3N degrees of freedom. You are aware that the molecule shows translational, rotational and vibrational motions. Since the molecule can move around anywhere in the three dimensional space, the position of the molecule can be specified in terms of the three coordinates of the centre of mass of the molecule. That is, we required three degrees of freedom to account for or to elaborate the translation motion. So, we are left with (3N - 3) degrees of freedom for rotational and vibrational motion.

The number of rotational modes depends on the geometry of the molecule. A linear molecule can rotate around only two of the three axes. If the molecule lies along the "x" axis, then it can rotate only around the "y" or "z" axis.  If we try to rotate the molecule around the "x" axis, then the positions of the atoms do not change.  We therefore say that a linear molecule can have only two rotational degrees of freedom along with its three translational modes. Therefore, a non-linear molecule might rotate around each of the three axes i.e., it has three rotational degrees of freedom. So we are left with (3N - 5) degrees of freedom for a linear molecule and (3N - 6) degrees of freedom for a non-linear molecule. Those are the vibrational modes of the molecule. Therefore, the linear CO₂ molecule will have 3(3) -   5 = 9 -   5 = 4 vibrational modes whereas, water    a non linear triatomic molecule    would have only 3(3) -   6 = 9 - 6 = 3 vibrational modes.