 So, when we think about the rotational motion of a quantum mechanical diatomic molecule, we've modeled that with the rigid rotor model, which tells us how to calculate the energy levels in terms of a rotational quantum number and this rotational constant, which is a collection of other constants, some of which, like mu and r squared, are properties of the molecule itself. That molecule, of course, that model, of course, was an approximation. When we set up the rigid rotor model, we connected two atoms with a bond length, r, and the whole purpose of the rigid rotor model is that that bond length is not changing. The r, the bond length r is constant in the context of the rigid rotor model. So this molecule is rotating, but r is not changing because the molecule is rigid. In fact, real-world molecules don't have rigid bond lengths. The bonds can change in length. That was the entire point of the harmonic oscillator model is to model changes in that bond length as the molecule vibrates. So it's more realistic to say those two atoms are connected by, I've connected them by a cartoon of a spring to indicate that the bond length can change, but if I stretch it, it experiences a force that pulls it backwards, and if I compress it, it experiences a force that causes it to expand. So that rotating molecule has a bond length that might be r, might be a little longer, might be a little shorter, depending on the vibrational state of the molecule. So that's a rotating non-rigid molecule. What that means is if we think about how that bond length will change as the molecule rotates, how that stretchy bond will interact with the fact that this molecule is rotating. As you know, if you've ever spun a flexible object, if I take a slinky and I spin it, it's going to elongate. That spring is going to stretch as the molecule rotates. The bond length is going to increase because of centrifugal effects, and the more rotational energy that molecule has, the greater the centrifugal effect, the more the increase will be. So as the rotational quantum number increases, as the rotational energy of the molecule increases, that increase in the bond length will get even larger. The effect of that rotational energy on the properties of the molecule, since r appears in the denominator of this rotational constant, when r is getting larger, that means the rotational constant is getting smaller as the molecule rotates, and even more so as it rotates faster. So what that means is the thing we've been calling the rotational constant isn't actually a rotational constant. So what do we need to do to improve our rigid rotor model? Let's write down some equations, but before we do that, since we've started talking about spectroscopy, it's easier to talk not in units of energies, but in units of wave numbers, so we can compare our results to spectra. So if I divide these energies by hc, that just turns my rotational constant into a rotational constant with units of wave numbers, and I did that of course by dividing by hc, so one of the h's in the numerator disappears, I should make it disappear, and if I divide by c, I'll just throw that c into the denominator. So that's our rotational constant in units of wave numbers. So rather than this equation, what we've just discovered is that the rotational constant or the rotational constant in units of wave numbers is going to be decreasing as l increases. So we could say that the rotational constant is some function of l that I need to then multiply by l and l plus 1. That's what we could do. We could figure out in what way does the rotational constant depend on l? It depends on r and what way does r depend on l. That's a little more complicated than we're actually going to bother to do, so that's a reasonable sounding approach, but it's not what we're going to do. Instead, we're going to take an approach that takes the same sort of thinking as we used for the anharmonic oscillator, which is to say our energy or our energy in units of wave numbers. Let's go ahead and use the rigid rotor approximation that we're used to, and then since we know that the rotational constant has decreased a little bit because of these centrifugal distortion effects, we'll subtract something from the rigid rotor energy, and the thing that we'll subtract is again some constant. We'll call that constant d. That's d for the centrifugal distortion, so this is going to be called the centrifugal distortion constant, and since we're doing this in units of wave numbers, we'll make sure that our constant has units of wave numbers, and we put a tilde on top of it. That constant, of course, is going to be different, that correction, of course, is going to be different for different l values, so we'll make sure and include some l dependence in this correction, so if we take a constant and multiply it, not by l and l plus 1, but that quantity squared, that's going to ensure that the size of this correction is increasing as l increases, and it also begins to make this look like a power series of the sort that we used for the harmonic oscillator, but now we're using rotational quantum numbers rather than vibrational quantum numbers. This is our empirical correction to the energy where the rough approximation is from the rigid rotor, and then an improved version comes from including this centrifugal distortion correction, so we can work an example and see how large an effect that is and how it works out, so if we stick with the carbon monoxide molecule for which we know the rotational constant is, I believe, 1.93, I'll give that to you with a few more sig figs this time, 1.9313 inverse centimeters, so that's a value in the microwave portion of the spectrum, which is not a surprise because we know rotational excitations happen in the microwave. The centrifugal distortion correction is much smaller, d sub e, that constant is 6 times 7 to the minus 6 inverse centimeters, so six orders of magnitude smaller roughly than the rotational constant. That looks ridiculously small. Turns out that the correction is not quite as small as that looks at first glance because let's just say we want to know the energy of the seventh rotational level, the energy of the l equals 7 rotational level, and I'll compute that in units of wave numbers. The reason I chose l equals 7 is because that's the most populated rotational level for a carbon monoxide molecule. Remember, at reasonable temperatures, the most populated level is not the ground state, but one of the upper states because of the degeneracy of those states. So to compute that energy, I need the b sub e times the rotational level and l plus 1, and the correction looks like a 7 and an 8 squared. So if we do these terms one at a time, 1.9313 inverse centimeters times 56 ends up significantly larger. That ends up being 108.15 inverse centimeters, and if I subtract from that, this correction, 6.12 times 7 minus 6 times 56 squared, now you see why the correction is not nearly as small as the constant itself. 56 squared has become a pretty large number, and so my correction, I didn't write it's roughly .02 inverse centimeters, so the net result is the centrifugal distortion correction has corrected my energy level for the seventh, l equals 7 energy level from 108.15 inverse centimeters, only corrected it down to 108.13 inverse centimeters. So in fact, the centrifugal distortion correction is very minor. We have to look out to the two digits past the decimal in units of wave numbers before I can see the size of this correction for the seventh level for carbon monoxide molecule. So it's a minor correction, maybe not quite as minor as the constant would lead you to believe, but it's not terribly important unless you're interested in frequencies all the way out to, in this case, out to the hundredths of a wave number. So there is one more non-ideality, one more correction that we're going to consider that turns out to be a little more important, and that's one that comes from the interaction between a bond that is not in fact rigid and a rotation that isn't in fact perfectly rigid rotor like as well. So that's what we'll consider next.