Given the energy of a diatomic molecule with masses m1 and m2 constrained to move along the x axis. The force constant of the bond connecting the two atoms is k.
E = ½ m1(x1 dot)2 + ½ m2(x2dot)2 + ½ K (x1-x2-xe)2
We make a change of coordinates from x1, x2, to X, x where the new coordinates are defined as
X= m1x1/(m1+m2) + m2x2/(m1+m2) and x= x1-x2-xe
X is the coordinate of the center of mass and x is the interatomic distance. After changing to the X,x coordinate system show that the energy reduces to that of the free translation particle of mass M where M = m1+m2 is the total mass of the molecule and the bound harmonic vibrational motion of a particle of mass µ= m1m2/(m1+m2) which is called the its reduced mass of the molecule.