To develop an understanding of, and ability to calculate, molecular energy levels. In diatomic molecules, there are two different ways that the molecule may move without its center of mass moving:
rotating around its center of mass and vibrating as if the two atoms are connected by a spring.
Energy may be added to the molecule by increasing the speed of rotation or the amplitude of the vibrations. As you should expect from quantum mechanics, energy must be added to molecules in specific quantities. Two of the solutions of the Schrödinger equation that you may have seen before--the hydrogen atom and the harmonic oscillator--will be useful in the study of molecules.
In looking at the Schrödinger equation for hydrogen, you learned that one important aspect of hydrogen is that it has a spherically symetric potential (i. e., the potential energy of the electron in a hydrogen atom depends only on its distance from the nucleus). This gives rise to the following equation for the allowed values of L2:
L2=l(l+1)ℏ2(l=0,1,2,3…),
where L is the angular momentum. When we look at the rotation of diatomic molecules, we also have a spherically symmetric potential energy function, specifically U(r)=0. Since this is the case, we can use the same equation for the angular momentum states that we used with hydrogen.
Compare the energy difference for adjacent vibrational states, which you just found, to the energy difference between rotational states, which you found in Part E. You should notice that vibrational states have much larger energy differences than rotational states. If you looked at the spectrum for a diatomic molecule, you would see sets of lines, with large gaps between the sets. Each set corresponds to a transition in vibrational state, whereas the lines in each set represent changes in rotational state accompanying the vibrational change. How much larger would the spacing between sets be compared to the spacing of lines within a set?