 Okay, so if we use what we understand about the energy levels for diatomic molecules that can both rotate and vibrate at the same time, we can understand quite a bit about the colors of light, the frequencies of light that those molecules will absorb. So the spectroscopy associated with row vibrational changes. So if we first think of a diatomic molecule as only behaving like a harmonic oscillator, just vibrating with no rotations, and we say the energies could be one half h nu, or three halves h nu, or five halves h nu, and so on, then we would predict this naive, overly naive prediction would be that it would absorb light with energy of h nu, or in units of wave numbers, nu should be with a tilde on top of it. So if we pull up a graph of what frequencies of light are actually absorbed by the carbon monoxide molecule, so this graph is for carbon monoxide, here's a spectrum of the infrared light that's absorbed by carbon monoxide. The actual fundamental vibrational frequency for carbon monoxide is about 2170 wave numbers. So the harmonic oscillator model predicts that light will be absorbed at wave numbers of 2170 inverse centimeters, and that's roughly what we see here, but you can see that it's not just absorption at this specific frequency, but it's a little more complicated than that. And the reason, so that there's several range of frequencies over which light is absorbed and you can see there's some structure in that absorption spectrum. So to understand where that comes from, we need to consider not just the vibrational energy, but also the rotational energies that are stacked on top of the vibrations to give us the row vibrational energy levels. And remember that the selection rules forbid this transition, we're actually not allowed to absorb just h nu, so we're not allowed to absorb exactly the frequency of the fundamental vibrational frequency. If we go from no rotational energy to a little extra rotational energy gaining one quantum of rotational energy from the L equals zero up to the L equals one state, then we'd absorb fundamental vibrational frequency plus two rotational constants. I've written them here in units of wave numbers. We could also go from the L equals one up to the L equals two. That would have going from two units of rotational energy up to six. So I've gained it four. So there's a bunch of different transitions I can make that gain rotational energy. Or if I start with L equals one and I lose, so I start with rotationally excited, but I end up not rotationally excited, then I've lost those two units of rotational energy so I can lose two units. Or if I start doubly excited and end up only singly excited, I go from six units, six multiples of the rotational constant to four, I'm sorry, six down to two. So then I've lost four multiples of the rotational constant. So you can see there's a bunch of different frequencies that the molecules should absorb. And if we zoom in on the spectrum, if I show you what this spectrum looks like zoomed in on this region around 2000 wave numbers, so we'll replace that graph with the next one. We can see that that is in fact what we see. There's a series of different frequencies at which the molecule absorbs light. There's one frequency in the middle of this diagram that looks like it's missing the hole in the middle of this diagram. That corresponds pretty closely to the fundamental vibrational frequency where the molecule is not allowed to absorb light by the selection rules. But if I increase that frequency by twice the rotational constant or four times the rotational constant or six, eight, ten, twelve evenly spaced lines in the spectrum on the positive side or evenly spaced lines in the spectrum on the negative side, fundamental vibrational frequencies plus multiples of the rotational constant or minus multiples of the rotational constant. So this would be where we'd expect the fundamental vibrational frequency to be and we understand why that one's missing. We also can understand why it has the shape that it has. Remember when we talked about populations of rotational energy levels. The low amounts of rotational energy, those levels are not very populated because they don't have a very high degeneracy. The populations get higher as we go to moderately, rotationally excited molecules because the degeneracy has increased. There's more ways for the molecule to exist in those states. But as we continue to go to even higher rotational excitations, the energy is increased to the point where the populations begin to drop. So each one of these lines would correspond, for example, let's see, the first line above the gap, that would be the L equals zero to L equals one transition. The next one would be L equals one to L equals two. Each one of those lines that we can identify with a particular rotational transition has a height proportional to the number of molecules that are in that initial state. So even the shape of the spectrum is a consequence of the population of those energy levels. There is one mystery, however, which is that for carbon monoxide, the fundamental vibrational frequency, as I mentioned, is 2,170 wave numbers. If we look more carefully at the diagram now, we see that the gap where I told you should be the vibrational frequency, that looks like it's a little below 2,150, more like 2,140 something wave numbers. So 2,170, that would be, I don't know, here somewhere. We don't quite understand why the gap in the middle of the spectrum that we think should come at the fundamental vibrational frequency is actually shifted down to lower frequencies, actually redshifted a little bit from the fundamental vibrational frequency. To understand why that's happening, we need to recognize that our model for these energy levels, the harmonic oscillator, rigid rotor model for the diatomic molecule, real diatomic molecules are of course not harmonic oscillators, they're not rigid rotors. So in order to understand the really small scale details of why the spectrum looks like it is, why we get this point at 2,140 some wave numbers rather than 2,170 wave numbers, we need to go a little beyond the harmonic oscillator and rigid rotor models. So we'll explore those ideas next.