 So we understand a lot of the features of the infrared absorption of diatomic molecules like carbon monoxide. But there's one more mystery about these infrared spectra that we will be able to explain. So we understand why the absorption has shifted, red shifted from the fundamental vibrational frequency for carbon monoxide of 2170, red shifted by a little bit to here. We understand why these additional rotational peaks layered on top of the fundamental vibrational frequency. If we zoom out though, I'm not looking at 2,000 wave numbers, but looking at a much broader range of wave numbers. So the peak we see here, the peak that's a little above 2,000 wave numbers, red shifted a little bit from 2170, that's absorption at almost the vibrational frequency, red shifted a little bit because of anharmonicity. If we look closely, however, you'll see that there's a small bit of absorption way up here a little above 4,000 wave numbers. And that we can't explain with the harmonic oscillator. That's again a feature of the anharmonicity of the molecule. So to start out, let's recall the selection rule for the harmonic oscillator. When we're changing vibrational levels for a harmonic oscillator, the selection rules tell us that the quantum number has to change by either plus one or minus one. I can go up one level on the energy ladder, or I can fall down one level, but that's all the harmonic oscillator is allowed to do if I'm using light to make the transition. Recall the reason for that selection rule was this transition dipole moment. So if I'm making a transition from N to N prime, calculate this transition dipole moment. That turned out to be zero when N and N prime differed by any value other than one. That in turn was because of the properties of the hermit polynomials that are hiding inside these wave functions. The hermit polynomials cause this function to disappear unless N and N prime differ by exactly one. That's all true for the harmonic oscillator. So if I draw a graph of the potential energy of a harmonic oscillator, so symmetric parabolic potential, the energy levels that show up at 1 half h nu or 3 halfs h nu or 5 halfs h nu and so on, the wave functions that occupy those energy levels have shapes that look like, I'll draw the wave functions in a different color. The ground state wave function looks like a Gaussian. The next wave function up looks like this Gaussian with a node in the middle of it. The reason for this cancellation comes because of the symmetry of these functions, the perfect symmetry, anti-symmetry of this function, and symmetry of this function causes when I include the X in the integral, if I take the product of this wave function and this wave function with an X, everything's fine, but if I do this wave function with a self or this wave function with the next one up, then the integral will cancel. That's all reminding you of what we had seen for the harmonic oscillator. Let's think about how things would be different for an anharmonic oscillator. If I redraw that same diagram for an anharmonic oscillator, the difference in the potential is asymmetric. It goes up to very large values for compressing the bond when I make the bond displacement shorter. It gets much softer and reaches a plateau value when I stretch the bond. The energy levels in this case, if I draw the ground state and one of the excited state energy levels, what the wave functions look like because the function is not symmetric anymore, I get a less symmetric down at the ground state. It's still fairly symmetric looking next state up because the function has been extended in the positive X side. The left or the negative lobe of this function begins to look unlike the right side or the positive lobe of this function. The perfect symmetry that existed, the perfect anti-symmetric or symmetric wave functions that we had for the harmonic oscillator, they're no longer perfectly symmetric. The cancellations that happened when I performed this integral for a harmonic oscillator are not going to be perfect cancellations for the anharmonic oscillator. The result of that is, let's say we have our normal selection rule, delta n is plus or minus 1 for the harmonic oscillator. That was allowed. If delta n is plus or minus 2, harmonic oscillator that was disallowed, delta n plus or minus 3, that's disallowed or forbidden. For the anharmonic oscillator, what we find when we do those integrals, certainly plus or minus 1 is still allowed. Those integrals are still non-zero for the same reason they were for the harmonic oscillator. So that means either perfectly harmonic or anharmonic oscillators can absorb energy at the fundamental vibrational frequency or slightly redshifted from that when they changed their vibrational state by 1. But when they changed their vibrational state by 2, harmonic oscillator wouldn't absorb any light. Anharmonic oscillator, the transition dipole moment integrals are not exactly zero, but they're much smaller than they would have been for an allowed transition. So we don't get zero, we get a small number. So what that means is absorption of light is not forbidden, it's just very weak. So that's the origin of this peak right here. You'll notice that these frequencies a little above 4,000 or double these frequencies of a little above 2,000. So this absorption comes at roughly twice the fundamental vibrational frequency. Again, redshifted by a little bit because of the anharmonicity, much weaker in intensity because the integrals are small, but they're not completely zero. So there is some weak absorption there. We could look a little further out at a little above 6,000. And in fact, if we zoomed in here, we'd say even weaker absorption out here at plus or minus three times the vibrational frequency. So that becomes quite a bit weaker. And these absorption at integer multiples of the vibrational frequency, so not new to be, but twice new to be or three times new to be, those weak absorptions at multiples of that frequency, those are called overtones. Again, out of an analogy with the musical situation, an overtone of a musical frequency is double the original frequency or triple the original frequency and so on. So same thing is true for quantum mechanical molecules. There's strong absorption at the fundamental frequency, weak absorption at the first overtone, which is double the original frequency, even weaker absorption out here at three times the original frequency. So that absorption of overtones is a feature of real world molecules that are not behaving perfectly as a harmonic oscillator because they're somewhat anharmonic. So that's a feature of the non-ideality of the harmonic, the vibrational properties of these diatomic molecules. Turns out that their rotational behavior is also not ideal as well. We've treated them as if they're rigid rotors with an idealized model. Turns out the non-idealities of the rigid rotor model also affect the properties of the absorption of these molecules as well. So we'll explore that next.