 So, putting together what we know about the harmonic oscillator model and the rigid rotor together, we know that a molecule can have both vibrational and rotational quantum numbers. It has vibrational energy and a rotational energy, which I've written here using the harmonic oscillator model, the rigid rotor model. If we wanted to, we could add corrections for the anharmonicity and corrections for the centrifugal distortion. So that would be a combination of an anharmonic oscillator and a centrifugally distorted rotor. But even that combination is still missing something, as we can see if we, again, remind ourselves of what these models are telling us about the basics of the motions of these molecules. The harmonic oscillator says, I have the molecule that's behaving as if it's attached by a harmonic spring. The bond length can be long sometimes or short sometimes, and it oscillates around some equilibrium position. Rigid rotor says the molecule is connected by a bond that has a particular length, and it rotates. I'm drawing these two pictures to point out that there's an inherent contradiction between these two models. The harmonic oscillator requires that the bond change length. The rigid rotor requires that the bond remain constant. So we've, in the centrifugal distortion correction, we talked about the fact that making the molecule rotate with higher rotational energy will cause the bond to stretch a little bit. But there's another contradiction in making the molecule vibrate, increasing its vibrational energy will also change its average bond length. So to see why that's true, let's draw a cartoon of the potential energy for a harmonic oscillator. So v of x, or if we prefer v of r for the bond length of this molecule. Because it's a perfectly symmetric harmonic oscillator, if it's in the ground or the first excited or any of the energy levels, the average bond length of the molecule is the same as the minimum of this, the bond length at the minimum of this well, this thing we've been calling r sub e. So the average bond length is the same. It stretches in equal distance on the positive x side as it compresses in the negative x side. That is not the case, however, for an an harmonic oscillator. So if the minimum of the well occurs at r sub e, a molecule on the ground or first excited or second excited energy level spends more time at significantly longer positive stretches than it does at the shorter compressed bond length. So what that means is the average bond length of the molecule is a little larger than r sub e. And that's going to be more true for the excited states, which are even more asymmetrical than it is for the lower states. It's going to be approximately true for the ground state, but the higher the energy level, the more distorted that bond length gets. That bond length is going to get larger than r sub e. So what that means is, as the vibrational quantum number increases, as they climb the vibrational ladder from ground state, n equals 0, up to n equals 1, n equals 2, and so on, as that vibrational state increases, the average bond length of the molecule increases. And because this rotational constant has an r in its denominator, h over 8 pi squared mu r squared times c, r shows up in the denominator. So as the average bond length is increasing, the rotational constant is decreasing. Notice that this decrease is for a different reason than the centrifugal distortion correction. Both involve the bond length increasing. The centrifugal distortion was the faster I make this molecule spin, the more rotational energy it has, the more its bond length stretches. This is somewhat different. The higher I excite the molecule vibrationally, the more vibrational energy the molecule has. The higher up this ladder I am, the larger the bond length is. So there's two different causes that are causing the rotational constant to decrease. This one is the change to the rotational constant caused by vibrational excitations. So that's why we call it rotational vibrational coupling. It's a coupling between the vibrational motion of the molecule and the rotational energy. So what do we do about that? This suggests that we need a different rotational constant for every different n level. So the way we end up implementing that is instead of using one single value b sub e that works for all vibrational and rotational states, we'll use a different rotational constant that we call b sub n that depends on the value of the vibrational constant. And the way we calculate this is we'll say the rotational constant in vibrational level n is the default one, b sub e that we're used to minus, the correction is negative because we know the rotational constant needs to decrease, minus another constant. This is the rotational vibrational coupling constant. So that constant multiplies the vibrational quantum number in the same form that it appears in the harmonic oscillator expression because we want this correction to get larger as the vibrational quantum number increases. So if I take that revised constant and plug it in instead of using the default b sub e, if I use b sub n instead, then what we end up with for our coupled rotational and vibrational energy, harmonic oscillator approximation for the vibrational energies, b sub e times l times l plus 1 would be the rotational energy if it were a rigid rotor. That's from this term, multiplying l times l plus 1, and then I have a correction, this alpha times n plus 1 half also multiplies l and l plus 1. And this is our rotational vibrational coupling correction. And again, notice that it involves both the vibrational quantum number n as well as the rotational quantum number l in this expression coupling together the level of vibrational and rotational excitation. So this is the rotational vibrational coupling correction that tells us how much the rotational energy gets reduced because of the excited vibrational energy. So that has only included the harmonic oscillator and rigid rotor with this new correction, the one we're calling the rotational vibrational coupling correction. What we'll do next is combine all these corrections back into one, taking into account anharmonicity and centrifugal distortion and the rotational vibrational coupling.