# Lennard-Jones potential and vibrational energy level diagram explanation

I don't understand how to interpret graphs like the following:

Forget the difference between the Harmonic and Morse (I think I get that), It's more what the horizontal lines mean and what the boundary line means. I know that vibrational energy is quantized and I can rationalize the shape of the Lennard-Jones potential but I don't understand how they're combined in this diagram. Please could you walk me through what's going on?

• Each horizontal line shows an interval of possible internuclear distances for a particular vibrational level: the higher you get in the ladder of vibrational levels, the bigger is the amplitude of vibration, and consequently, the bigger is the corresponding interval of possible internuclear distances. Apr 6, 2015 at 13:32
• And, of course, as user1420303 correctly pointed out in his answer all of what I said in my comment above has to be taken as "speaking classically", i.e. in terms of classical physics. In reality, there is a non-zero probability for nuclei to be at a distance bigger than one allowed classically. Apr 7, 2015 at 8:04

## 1 Answer

This diagram is conceptually horrible, because it is a mix of quantum and pseudo-classical physics.

Forget for a while the vibrations. If you calculate single point energies (within Born-Oppenheimer approximation) for different internuclear values, you'll get something like blue curve.

Now, for the $r_e$ distance between nucleus (again, within Born-Oppenheimer approximation) you can calculate the "vibrational (normally harmonic) frequency" in a pseudo-classic fashion. For this frequency, you can assign energies (of a quantum harmonic oscillator having this frequency). This energy values correspond to the difference between energy values for the horizontal blue lines, and the well (blue).

The widths of the horizontal lines have not physical meaning.

For the sake of completeness, the $D_e$ would be the energy of equilibrium state if no vibrations are considered. $D_0$ when the fundamental vibrational state energy is add. The former energy is not measurable because this is not a real energy for the system.

• I agree that the diagram is indeed horrible. I think one should always plot probability densities for each vibrational level on a diagram of such kind (as in the corresponding Wikipedia article for example) to make it less horrible. :D Apr 7, 2015 at 8:09