1. Simple case: a diatomic with masses m1 and m2

2. Appropriate factorization:
Solutions (i.e. eigenvalues of F)
12 = 0; 22 =
= K/
Eigenvectors

3. In actual displacements:
Eigenvector 1: 2 = 0   
In actual displacements:
x1 = x2
translation of the molecule!

4. Eigenvector 2: 2 = K/    x1 m1 = - x2 m2  vibration!
COM stays at rest!

5. The ajk define a new set of coordinates, the normal coordinates:
Mode 1: Choose A1 = 1  A2 =
(because ) 
Normal mode:

6. Mode 2:
Choose A1 = 1  A2 = -
(because ) 
Normal mode: