The Morse potential, named after physicist Philip M. Morse, is a convenient interatomic interaction model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the quantum harmonic oscillator because it explicitly includes the effects of bond breaking, such as the existence of unbound states. It also accounts for the anharmonicity of real bonds and the non-zero transition probability for overtone and combination bands. The Morse potential can also be used to model other interactions such as the interaction between an atom and a surface. Due to its simplicity (only three fitting parameters), it is not used in modern spectroscopy. However, its mathematical form inspired the MLR (Morse/Long-range) potential, which is the most popular potential energy function used for fitting spectroscopic data.

The Morse potential energy function is of the form

Here r is the distance between the atoms, re is the equilibrium bond distance, De is the well depth (defined relative to the dissociated atoms), and controls the 'width' of the potential (the smaller is, the larger the well), The vibrational structure in the Morse Energy potential is given by:

An experimentalist has measured the following parameters for a diatomic molecule and wanted to create Morse potential curves and all the populated vibrational states for his data:

Calculate the dissociation Energy () for each electronic states

Using your math software, create a table of distances between atoms starting from 1.000 Angstrom to 5.000 Angstrom with a step of 0.004 Angstrom

For each value of distances, calculate Morse potential Energy for the Ground and Excited electronic states and create a table for the values

Plot calculated Morse potentials against the distances

Calculate the first 4 low lying vibrational states in each of the elecronic states.

Show these states in the Morse potential plot by adding as new traces

Assuming only two vibrational states are populated in the ground electronic states and all of the four vibrations can be populated in the excited state, how many peaks he would observe when he measured the grouund to excited state transition?

Make a table and list all the electronic+vibrational energies for the transitions. (Assume there is no forbidden transition)

What would be the spacing between each observed vibrational structure in this excitation? (Hint: subtract the lowest Energy transition from all transitions and relist the table in the f by using relative values)