The potential energy associated with small bond stretches about the equilibrium bond length R_e can be approximated by a parabolic equation.

where k is the force constant in (J/m²) that gives the stiffness of the bond, and x in (m) that gives the amount of bond stretch.

The Schrödinger's equation in SI units for the motion of two atoms of masses m₁ and m₂ in respects to each other is:

Because the vibrations are about the center of mass the effective mass becomes the reduced mass m , defined by the following equation:

The permitted vibrational energy levels for a diatomic molecule undergoing simple harmonic motion are:

Where w is the classical vibration frequency in radians per second (divide by 2p to convert to cycles per second) and n is the vibrational quantum number. The analytical solution can be expressed in terms of hermite (h_n(x)) polynomials:

The first two solutions to the harmonic potential for diatomic hydrogen are given at the top of this page. There are several noteworthy aspects of these solutions.

• First, they look very much like the particle in a box solutions except that they tunnel slightly into the potential. The exponetial term in the analytical solution describes this tunneling.
• Second, the zero point energy is not zero. It is impossible to freeze the vibrational motion out of a molecule. All bonds will have and intrinsic vibration energy.
• An finally, the energy spacing between each different eigen state is a constant. The harmonic potential increases at exactly the right rate to insure constant steps in the total energy. Since IR spectroscopy measures the energy difference between vibrational states, there will be only one characteristic frequency measured for the vibration of a diatomic molecule.