# Number of vibrational modes for a molecule

All atoms in a molecule are constantly in motion while the entire molecule experiences constant translational and rotational motion. A diatomic molecule contains only a single motion. Polyatomic molecules have more than one type of vibration, known as normal modes.

### Molecular Vibration

A molecule has translational and rotational motion as a whole while each atom has it's own motion. The vibrational modes can be IR or Raman active. For a mode to be observed in the IR spectrum, changes must occur in the permanent dipole (i.e. not diatomic molecules). Diatomic molecules are observed in the Raman spectra but not in the IR spectra. This is due to the fact that diatomic molecules have one band and no permanent dipole, and therefore one single vibration. An example of this would be O_{2} or N_{2}. However, unsymmetric diatomic molecules (i.e. CN) do absorb in the IR spectra. Polyatomic molecules undergo more complex vibrations that can be summed or resolved into normal modes of vibration.

The normal modes of vibration are: asymmetric, symmetric, wagging, twisting, scissoring, and rocking for polyatomic molecules.

Symmetricical Stretching |
Asymmetrical Stretching |
Wagging |

Twisting | Scissoring |
Rocking |

**Figure 1:** Six types of Vibrational Modes. Taken from publisher http://en.wikipedia.org/wiki/Infrared_spectroscopy with permission from copyright holder.

### Calculate Number of Vibrational Modes

3*n* degrees of freedom describe the motion of a molecule in relation to the coordinates (x,y,z). The 3*n* degrees of freedom also describe the translational, rotational, and vibrational motions of the molecule. There are three degrees of freedom for translational, movement through space, and rotational motion, each for a nonlinear molecule. Therefore, translational and rotational can move and rotate around each of the three Cartesian axes. However, a nonlinear molecule can only rotate around 2 of the Cartesian axes because the rotation about the molecular axis does not represent a change of the nuclear coordinates. If you subtract the translational and rotational degrees of freedom, you obtain the following equations shown below for the degrees of vibrational freedom.

The degrees of vibrational modes for **linear molecules** can be calculated using the formula:

\[3n-5 \tag{1}\]

The degrees of freedom for **nonlinear molecules **can be calculated using the formula:

\[3n-6 \tag{2}\]

\(n\) is equal to the number of atoms within the molecule of interest. The following procedure should be followed when trying to calculate the number of vibrational modes:

- Determine if the molecule is linear or nonlinear (i.e. Draw out molecule using VSEPR). If linear, use Equation 1. If nonlinear, use Equation 2
- Calculate how many atoms are in your molecule. This is your
*n*value. - Plug in your \(n\) value and solve.

Example 1: \(CS_2\)

An example of a linear molecule would be \(CS_2\). There are a total of \(3\) atoms in this molecule. Therefore, to calculate the number of vibrational modes, it would be 3(3)-5 = 4 vibrational modes.

Example 2: \(CCl_4\)

CH_{4} is an example of a nonlinear molecule. In this molecule, there are a total of 5 atoms. Therefore, there are 3(5)-6 = 9 vibrational modes.

Example 3: \(POCl_3\)

A more complex example could be \(POCl_3\). The shape of this molecule dictates that this is a nonlinear molecule. It contains 5 atoms and therefore would have 9 degrees of vibrational freedom.

Why would CO_{2} and SO_{2} have a different number for degrees of vibrational freedom? Following the procedure above, it is clear that CO_{2} is a linear molecule while SO_{2} is nonlinear. SO_{2} contains a lone pair which causes the molecule to be bent in shape, whereas, CO_{2} has no lone pairs. It is key to have an understanding of how the molecule is shaped. Therefore, CO_{2} has 4 vibrational modes and SO_{2} has 3 modes of freedom.

### References

- Harris, Daniel C., and Michael D. Bertolucci.
*Symmetry and Spectroscopy: an Introduction to Vibrational and Electronic Spectroscopy*. New York: Dover Publications, 1989. Print. - Housecroft, Catherine E., and Alan G. Sharpe.
*Inorganic Chemistry*. Harlow: Pearson Education, 2008. Print. - http://en.wikipedia.org/wiki/Infrared_spectroscopy