An atom moving in three dimensions has three degrees of freedom corresponding to the freedom in movement in, say, the x, y and z directions.  A collection of N unabound atoms will have  3N degrees of freedom.  If the N atoms are bound through the formation of a molecule, the 3N degrees of freedom get redistributed into translational rotational and vibrational modes.  Since the molecule can be translated as a unit, there are three translational modes (degrees of freedom).  Similarly there are three rotational modes with respect to three independent axis of rotation.  The remaining, 3N-6 are the vibrational modes.  For a linear molecule, since there are only two rotational modes with respect to the two axes perpendicular to the molecular axis, there are 3N-5 vibrational modes.

          If the potentials energy functions for all the motions can be assumed to be harmonic, then the 3N-6 modes can be categorized into 3N-6 normal modes. Consider the example of water.  There are three atoms and 3N-6 = 3 normal modes.  In terms of the potential energy functions for vibrations, there are three functions: one each corresponding to each O-H bond and one corresponding to the H-O-H bending.  In terms of the individual bond vibrations, the vibrational motion can appear quite complex.  The total potential energy P.E. may be written as:

P.E. = ½ k (r₁-r₁₀)² + ½ k (r₂ – r₂₀)² + ½ k^′(θ-θ₀)²                                                                            (13.26)

Here, r₁₀  and    r₂₀ are the equilibrium bond lengths of the two O-H bonds and θ₀ is the equilibrium bond angle.  A normal mode of vibration is defined as a vibration in which all atoms oscillate with the same frequency and pass through their equilibrium positions at the same time.  The center of mass is unchanged during a normal mode.  The three normal modes of vibration of water are shown in the following figure.