 So, so far, in trying to understand how diatomic molecules behave, we've considered two different components of their motion. Their translational motion as their center of mass moves around in the container that they're contained in, we've described with the particle in a box model. That model we've used to describe the coordinates x, y, z, so let me go ahead and draw a diatomic molecule. Its center of mass is somewhere, that center of mass has some x, y, z coordinates. The rotation of the molecule, as the molecule rotates in this direction or this direction or some other direction, we've used theta and phi to describe those coordinates. So the molecule, we've allowed it to move around in x, y, and z, we've allowed it to rotate in theta and phi. What we haven't allowed it to do or what we haven't examined so far is changes in the bond length of the molecule. So the bond length is the variable we need to use to describe the vibration of the molecule. As the molecule's bond length changes, we call that motion a vibration and we need to understand how to quantum mechanically describe the states of the molecule and eventually the thermodynamic properties of the molecule due to those vibrations. So that's the next thing we'll tackle. So we're going to back up all the way to quantum mechanics, solve Schrodinger's equation again, not for the particle in the box to describe the motion of the molecule, not for the rigid rotor to describe the orientation of the molecule, but for the bond length to describe these vibrational motions. So the first step that we always need to do in any quantum mechanical problem is to write down the potential energy. We can't write down Schrodinger's equation until we know what potential energy to plug into that Schrodinger equation. So the potential energy is going to depend on this variable R, the bond length. Remember that in the previous two approaches, both for particle in the box and rigid rotor, we started out by saying, well, it may not be exactly true, but let's say the potential energy is equal to zero. Can we do that for the vibration of a molecule? We definitely can't. If you think about how the potential energy of the molecule changes as I change its bond length, so if I have a diatomic molecule, if I stretch it, or if I compress the bond, thinking about what happens to the potential energy of the molecule is definitely going to change. Unlike rotation where I shouldn't really expect the energy of the molecule to be different if I just place the molecule at a different orientation or translation if I just move it to a different position in the box. If I draw a qualitative picture of what that potential energy should look like as I change the bond length. So I've got a molecule, it's at some bond length. If I stretch the bond, I'm going to have to put energy into the molecule to stretch the bond. I'm raising the energy of the molecule when I stretch the bond. Likewise, if I compress it below the bond length that would naturally tend to have, that's also going to require some energy. So there's some energy where there's some bond length, I should say, where the energy of the molecule is lowest. And if I increase the bond length beyond that, or if I decrease the bond length below that, the energy is going to go up from that point. So there's some minimum energy, I'll draw it in approximately the right place. I'll call that V sub e. R sub e, the e stands for equilibrium. The equilibrium bond length, if I just let the molecule alone, the bond length that naturally tends to have is this equilibrium bond length. That corresponds to the lowest energy. And the energy goes up as I stretch the bond. It doesn't keep going up forever because if I keep stretching the bond, eventually the molecule will break. And then if I have two atoms separated by a long distance, if I move them further apart, I'm not changing their energy anymore. So this energy will go up and approach zero as I've drawn the graph. Eventually a very long bond length. It doesn't do the same thing on the other side because if I think about what happens when I compress the bond, got molecules that they're equal to bring in bond length, if I move them closer, that raises their energy. If I move them closer and closer still, eventually I'm going to start overlapping the two atoms. And it becomes very energetically difficult to overlap the two atoms. So as the energy approaches, as the bond length approaches zero, the energy is going to begin to rise very rapidly and become very large. So this is a general shape qualitatively anyway for what the potential energy should look like. And it's not a very good approximation to say that I can approximate that curve with just a flat v equals zero curve. So we're going to have to do a little bit more work to do the quantum mechanics this time. We'll have to figure out what this potential energy curve looks like and how to insert it into Schrodinger's equation. The bad news, there's bad news of a couple different flavors. The first piece of bad news is it's not actually known. I can't write down an equation for what this curve actually looks like for real world molecules. If I could, it would be quite a complicated expression. And if I insert it into Schrodinger's equation, Schrodinger's equation is going to be very difficult to solve. Certainly on blackboard without using very powerful computers. So what we'll do is what physical chemists often do when we're presented with a problem that they can't solve. I don't know the right equation and even if I did, I couldn't solve it. So what a physical chemist often does in this circumstances is to come up with a model, an approximation, a simplification. So essentially we're saying I can't solve this problem. Let me think of a simpler problem that I will be able to solve. And that simpler problem, if I can't solve the problem for this green curve that I've just drawn, I'm going to draw a simpler curve. That this curve reminds me of, a little bit ahead of myself. But the curve I've just drawn here is a parabola. So as the distance away from r sub e increases, let me make it increase symmetrically. And if I make its increase as the square of the distance away from r sub e. Then what I've drawn is a parabola with some constant out front. So this expression that I've just drawn for the potential energy is a simpler equation. And it turns out when we use that for the potential energy in Schrodinger's equation, we will be able to solve Schrodinger's equation. It's not terribly different from this expression, especially not down at the bottom and we'll see as we go along that usually for diatomic molecules, they spend most of their time down here near the equilibrium bond length. So this approximation is not actually too bad. We will simplify this expression a little bit by saying, let's let instead of writing this expression as the difference between the bond length and the equilibrium bond length, let's define a new variable x to be this quantity r minus r sub e, the amount by which I've stretched the bond. When x is a positive number, I've stretched the bond to a larger bond length than the equilibrium bond length. When x is negative, I've compressed the bond to less than its equilibrium bond length. So this quantity is called the bond displacement. How much I've displaced the bond away from its equilibrium bond length. And with that change, I have a potential energy that depends not on bond length, but now on this new variable x, the bond displacement that I've just defined. So that's the expression that we'll use and insert into the Schrodinger equation to solve for the quantum mechanical behavior of vibration of these diatomic molecules. That model is called the harmonic oscillator model. And that name may sound a little bit cryptic, it's not obvious necessarily what's harmonic or what's oscillating in this particular model. You might think we would call it the vibrational model or something like that. So the next step will be to explain a little more about this model and why it is called the harmonic oscillator model.