The vibrational energy of a diatomic molecule consists of a kinetic energy component and a potential energy component:

In classical mechanics, the potential energy is equal to the work done against a force in moving a body from the reference point, where , to the displacement . In other words,

According to Hooke’s law, the vibrating diatomic molecule experiences a restoring force that is proportional to the displacement between the two atoms relative to its equilibrium or undistorted length:

where is the force constant. Substitute eq55 in eq54 and integrating, we have

Substitute eq56 in eq53

From the theory of conservation of energy, the average kinetic energy of the molecule undergoing simple harmonic vibrational motion equals to the average potential energy of that molecule (see diagram above), i.e. . Hence, for a system of molecules, the average vibration energy of a molecule is:

According to the equipartition theorem mentioned in an earlier article, each kinetic energy component of a molecule has an average energy of . Hence, a diatomic molecule has an average vibrational energy of (vibration is along one axis).

For a molecule with atoms, we can derive its total component of motion as follows:

The motion of an atom is described by three Cartesian coordinates: , and . The components of the motion of independently moving atoms are therefore . However, if these free atoms are bound together to form a molecule, the translational motion of the molecule has three degrees of freedom regardless of its structure (see this article). The molecule’s rotational motion has two degrees of freedom if it is linear and three degrees of freedom if it is non-linear. The remaining degrees of freedom are attributed to the vibrational motion of the -atom molecule. Therefore, we have: