 So we've seen the mystery that when we try to understand the row vibrational spectrum of diatomic molecules, like this one shown for carbon monoxide, even though we think we understand many of the details of this spectrum, the fundamental vibrational frequency for carbon monoxide is 2170 wave numbers, and we think that would show up right where the gap appears in the middle of the spectrum, but that gap actually shows up at a value that's different than 2170 wave numbers. So we need to understand why that happens. The reason that happens turns out to be because of the non-harmonicity, the anharmonicity of the carbon monoxide molecule. So carbon monoxide molecules, diatomic molecules in general, actual real molecules, are not perfectly harmonic oscillators. They're somewhat anharmonic. So remember when we first started talking about the harmonic oscillator, this curve is what the potential energy looks like for a real molecule. There's some bond length that we call the equilibrium bond length where the energy is lowest, and then the energy rises on either side of that. That problem was a little too difficult for us to want the challenge to insert that equation into Schrodinger's equation. So we took a simpler approach and said we'll pretend that the energy instead looks like a parabola. So the pink curve that I've just drawn here is the one that we actually inserted into the Schrodinger equation and is an approximation for the real curve that works pretty well at low energies, but it begins to work less and less well as the energy increases. But we're not always interested in the energy at the bottom of the curve. In fact, we know that the harmonic oscillator has a zero point energy. So the lowest energy level might be here, might be somewhere in this curve. So for the lowest energy level, it might work okay. The next energy level up, the next energy level up, the higher we climb in this well, the worse the estimates of the energies get when we use the harmonic oscillator model to predict the energy because the real potential energy curve becomes more and more different as we climb that well. In particular, this K in the harmonic oscillator model that described the curvature of this parabola, a large value of K would be a steeper curve, a small value of K would be a softer, looser curve. You can see that as the energies go up, the real curve, this green curve, the real potential energy curve is getting softer and softer and softer. So as we go to higher energy levels, we might want a somewhat shallower parabola or a somewhat shallower one still. So the further up this energy well we go, the smaller the value of K should go. So in other words, as the energy levels increase, the value of K that we should be using as our harmonic oscillator approximation to this well is going to decrease. And since the vibrational frequency that we're using to predict the spectrum of the molecule, that depends on K. So as K is dropping, that vibrational energy, that vibrational frequency would be dropping as well. So actually our harmonic oscillator prediction for the vibrational frequency is going to be off by a little bit and it's going to be off by more for the upper energy levels than the lower energy levels. So the next question is how do we go about coming up with a better approximation than the harmonic oscillator approximation to make better predictions of the experimental properties of carbon monoxide and other diatomic molecules? We can consider several approaches. One is we could use the correct, the actual potential energy function. If I could write down for you the actual potential energy function of this green curve as opposed to the pink approximate harmonic oscillator approximation, we could plug that into Schrodinger's equation, solve Schrodinger's equation, come up with new way functions, energies, repeat all the work we've already done. The bad news or good news, if that doesn't sound like fun, is that we can't do this. I can't write down for you what the correct potential energy function is. And even if I could, we wouldn't be able to solve Schrodinger's equation for it, at least not in a simple form on the blackboard. So the second approach we can consider taking is coming up with some more approximate models, some additional approximate models for the potential energy function. After all, that's what we've done so far. We said the real problem is too hard to solve. Let's solve this simpler problem instead. Let's come up with a model that maybe isn't quite as bad as the harmonic oscillator is a little more accurate. But for example, we could say the potential energy, instead of saying remember where this one-half kx squared came from, it was some constant term, some linear term that disappeared, some quadratic term. If I continue that power series, I could say there's an additional term, I'll just call it k prime, that might depend on the bond displacement cubed. And I can continue that series as long as I want. So that looks like a better approximation of the potential energy function, even if I don't know the real answer. The bad news for that function would be so if if the quadratic function looks like this pink curve, a cubic function, once I include a cubic term, that's going to look like, remember what cubic functions look like, it's going to look like this. So it's going to go up in this direction, but it's going to go down in that direction. That might indeed do a better job of approximating the real function, it gets softer and softer as it goes up, but eventually it's going to turn around and dive down to negative infinity. So that's maybe okay in this region, but it's going to give us some energy levels over here that are complete nonsense. And in fact, those energy levels are going to be lower than the energy levels in this well. So that has some problems that approach. That doesn't mean we can't come up with some other approximation. We could say we could come up with some equations that aren't polynomial power series equations. In fact, one very common and not terribly complicated equation that can be used is this one that's called the Morse potential. So this potential energy function in fact does have the correct shape. So potential energy as a function of position, this one minus an exponential squared multiplied by some constant that represents the amount of energy it takes to break that covalent bond. That function looks an awful lot like the correct function and you can do a pretty good job of modeling the covalent bond in a harmonic oscillator or an anharmonic oscillator with this Morse potential. You can actually plug it into Schrodinger's equation, solve Schrodinger's equation, come up with wave functions and energies and they are reasonable approximations. Again, that's a little more involved that we're then we're going to get it in this particular course. So instead, we're going to take a a third approach, which is instead of using an approximate potential energy and continuing to do quantum mechanics, we're going to take a purely empirical approach. Forget about quantum mechanics. Take what we've learned from the quantum mechanics we were able to do, which is that the energy levels are evenly spaced, 1 half h nu or 3 halves h nu or 5 halves h nu. That's what the harmonic oscillator model told us. As a way of building in and harmonicity, we'll just take that general trend that the higher we go in this energy level, the higher we go in this energy well, the softer the potential energy function becomes, the smaller the force constant becomes, the lower the vibrational frequency gets. So instead of multiplying by the same vibrational frequency, we can think of it as multiplying by a number that keeps decreasing as the value of n goes up, or we could do the following. We could say, let's use this as our initial approximation, n plus a half h nu, and then let's subtract something from it. We know that those energies should decrease. The correct energy level should be a little below the prediction that the harmonic oscillator made. So if I draw that picture in a little more detail, so instead of 1 half h nu, 3 halves h nu, 5 halves h nu, we want each one of those to be shifted down by a little bit. And we want that correction to be larger for the upper levels, moderate and smaller as we go down the ladder. So these energy levels are harmonic oscillator or an harmonic oscillator. So we want a correction that's changing in size as we go up the ladder. One way to do that would be to have, take our quantum number n plus a half, subtract not n plus a half, but n plus a half squared times h nu multiplied by a correction factor, x sub e. So this correction factor, we call the anharmonicity correction or the anharmonicity factor. And that's a purely empirical, purely experimental measurement based correction to the observed energy. So what we would do is say quantum mechanics told us that, or at least the harmonic oscillator model plus quantum mechanics told us that this point should come at 2170 wave numbers. In fact, we see it at 2143 wave numbers. That suggests that the size of this correction we need to make in order for the energy levels to have the values that we observe them to have. And for this carbon monoxide molecule in particular, if I give you the value of x sub e, that's a pretty small correction. It's 0.0061. There's some a little bit of advantage in thinking of the value of x sub e, because you'll notice that this term h nu and this term h nu, those are both a quantum of vibrational excitation. And this is just telling me essentially a fractional term for how much I'm correcting the harmonic oscillator model to build in the anharmonicity. So this tells me that that anharmonicity correction is down in the vicinity of roughly a percent or so. More commonly, however, we quote the value not of x directly, but x times the vibrational frequency. And if we're working in wave numbers, that anharmonicity correction multiplied by the fundamental vibrational frequency in units of wave numbers, that works out to be roughly 13 wave numbers. So we can see that when we throw in the anharmonicity correction to try to correct for the errors in the harmonic oscillator model, that correction is on the order of roughly 10 or so wave numbers as corrections on top of this 2170 wave numbers. And if I include various factors of n, that ends up correcting these energies from 2170 down to a few multiples of 10, 2143 wave numbers. So this empirical approach is very commonly the approach that's used to treat anharmonicity in practical real-world diatomic molecules. You've made enough measurements of the properties of the molecules. You don't have to, it's best not to treat them with the harmonic oscillator model, because that ends up making some errors. And this anharmonicity correction ends up being able to give you the correct predictions for the spectroscopy of those molecules.