 So now we're in a position to do significantly better than just the harmonic oscillator and rigid rotor approximation for describing diatomic molecules. Real diatomic molecules aren't perfectly harmonic and aren't rigid, so we know now we can write the energy of a particular vibrational state and rotational state using not just the harmonic oscillator approximation, which gives us this n plus one-a-half h nu or n plus a-half vibrational frequency in wave numbers, but with a correction for anharmonicity, the rotational energy looks like the, sorry that should be a plus, plus the rotational energy that we get from the rigid rotor minus a correction that comes from the centrifugal distortion and l and l plus one both squared, and we also have a correction that comes from the coupling between the vibrational and the rotational motions. So with those three corrections, anharmonicity, centrifugal distortion, rotational vibrational coupling, we can do much better than the original harmonic and rigid rotor models in describing the spectroscopy for diatomic molecules in the real world. So if I bring up a picture that we've seen before for the spectrum for carbon monoxide, we can understand now just about every feature of this and using these various corrections if we know the value of alpha and d and x sub e nu sub e, then we can predict the positions of each of these lines to usually an accuracy of about a hundredth of a wave number or even better. In particular, we understand why we've gotten redshifted from 2170 wave numbers down to a lower value for the center of the spectrum. We understand why there's rotational peaks stacked on top of these vibrational transition. We've talked about how there's a weak signal here from the excited state transitions. If we remind ourselves of the difference between these energy levels for the harmonic oscillator, so treating just the harmonic oscillator at first, so we have energy levels of one half h nu, three halves h nu for the anharmonic oscillator, which I'll draw in a different color. Those energy levels are shifted down slightly for the ground state, down a little bit more for the first excited state, down even more for the upper states, so they're shifted downwards. That's the cause for this redshift. If I layer the rotational states on top of these, so this is one quantum of vibrational excitation with no rotational excitation, if I layer a sequence of states with some rotational energy on top of the vibrational energy, so here's the ground vibrational states with one excitation rotationally or more. In the real molecule, just like the anharmonicity reduced the vibrational energy, the centrifugal distortion reduces the rotational energy and the rotational and vibrational coupling reduces these energy levels even more, so these don't get layered on in exactly the same way for the real molecule, and this is difficult to see from the scale of my drawing, but this range of rotational energies gets compressed by a little bit for the real molecule. That means two things. That means not only do we get a redshift in these energies when we excite vibrationally, but also when we excite rotationally, those rotational excitations are lower than we would predict. When it comes to looking at the details of this spectrum, that makes things in fact a little more complicated because if I remember the selection rules tell us I have to increase by one or perhaps decrease by one in the rotational level, likewise for the rotational level. Increase by one for the vibrational, also for the rotational level, I can also decrease by one for the rotational level. If I'm starting at a particular level, I can either go one higher or one lower when I do a rotational excitation in addition to my vibrational excitation. What that means is that when I increase by one vibrational excitation plus some, that means I've increased in rotational level and I've increased by less than the rigid rotor model tells me I should. This spacing between each of these peaks in this comb spectrum, we'd expect those to be twice the rotational constant. It's actually twice the rotational constant minus a little bit, minus a term that involves some corrections due to centrifugal distortion and real vibrational coupling. On the other hand, the absorption peaks that look like the fundamental vibrational frequency redshifted minus a little bit. That's because as I increase in vibrational energy, I've decreased in rotational energy. The result there is that these corrections to the energies are larger in the lower state than they are in the upper state. What that means is the gap between these peaks, instead of being exactly twice the rotational constant, it's twice the rotational constant plus a little bit. Now, if you look closely at this spectrum, you can probably see that. Over here, the bars are more finely spaced than they are over here. In fact, the further I go into the high frequency portions of the spectrum, the narrower those gaps get, the further I go into the low frequency part of the spectrum, the larger those gaps get. This comb spectrum is in a perfectly evenly spaced comb with every peak exactly twice the rotational constant above the previous one, but that spacing is changing slightly with corrections due to the rotational corrections in this molecule. Like I've said, if we know the values of these, not just the vibrational frequency and the rotational constant, but if we know the values of these corrections, these constants that correct for various non-idealities as well, then we can predict each of the positions of each of these peaks, we can predict exactly the frequencies that a diatomic molecule like carbon monoxide will absorb up to quite a few significant figures. The next thing we've talked about now that we fully understand diatomic molecules out to many, many sig figs, we can predict lots of features of their spectroscopy. Next question is to ask what we can understand about polyatomics. Can we get larger than diatomic molecules and understand something about their spectroscopy?