The Morse Potential is an empirical potential that describes the stretching of a chemical bond. It is asymmetric indicating that it is harder to compress a bond than to pull it apart.

where D_e is the depth of the potential minimum, k the bond force constant, R the distance between the nuclei of bonded atoms, and R_e the equilibrium inter nuclear distance.

It allows the calculation of a wide range of energy levels because it resembles closely the true potential. The permitted energy levels are usually calculated using perturbation theory as power series corrections to the harmonic oscillator energy. These energies are given by the following:

where w_e is the harmonic vibrational frequency and x_ew _eis the anharmonicity constant. The first anharmonicity constant is given approximately by:

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

• First, they are almost identical to the Harmonic oscillator solutions
• Second, They are displaced from zero. The average displacement <x> is not equal to zero as it is in the harmonic oscillator. In fact, as the energy increases the displacement becomes larger. The chemical bond lengthens as it is placed in increasingly higher vibrational states.
• An finally, the energy spacing between each different eigen state is no longer purely constant. The effect of the anharmonicity constant is to make the spacing between energy levels decrease as the vibrational quantum number increases.