 So, we know that for a harmonic oscillator, the energy levels are equally spaced. We know how to calculate the energy of each of these individual energy levels. We've begun to talk about the frequencies of light used to make transitions between these various different energy levels, but that leaves the question of which of these transitions is allowed. Am I allowed, for example, to make a transition from the 0th to the 1st or the 1st from the 2nd or the 0th to the 2nd? Which of these transitions is allowed or acceptable and which ones are not? So, when we find the answer to that, we'll know what are called the selection rules for which of these transitions are allowed. And remember, the way we determine which transitions are allowed is by using the transition dipole moment for the molecules, which is proportional to this quantity. If I want to make a transition from, let's say, the nth to the n prime energy level, easier to e1, easier to e2, from some initial n to some different n for the final state, then what I need to know is the integral of a wave function for the initial state and a wave function for the final state sandwiched around the dipole moment of the molecule in the direction, the component of the dipole moment in the direction parallel to the oscillation of the light that we're shining on the molecule. So, for a molecule like HCl that has some dipole moment that points in that direction because it has a positive side and a negative side, we might be interested in that dipole moment or depending on what direction the light is pointing in, the component of the dipole moment that's in the z direction. So, let's think about what this integral would equal. As a first attempt, we can say, well, if the dipole moment happens to be constant, if the dipole moment of HCl is just a number, two or three to buy whatever the dipole moment of that molecule is, if I pull that constant out of the integral and as a spoiler alert, this is going to be not the correct thing to do. So, if I've raised any eyebrows by pulling that dipole moment out of the integral, you're correct to be skeptical at this point. But if I do do that and I pull the dipole moment out of the integral as a constant and then ask what is this integral, that integral we know how to do. Before we even plug wave functions in and compute the integral, what is the integral of one wave function multiplied by a different wave function, a harmonic oscillator wave function multiplied by a different harmonic oscillator wave function, one of which has been complex conjugated. And that is guaranteed to be zero because those wave functions are orthogonal to one another. Every pair of wave functions, zero and one, zero and two, one and two, all those wave functions are mutually orthogonal. So this integral is guaranteed to come out to be equal to zero. So the whole integral must be zero. That suggests that none of these transitions are allowed. We can't use light to excite vibrational transitions in molecules. And we know that's not true. We know that infrared light is absorbed by diatomic molecules to excite some of these vibrational transitions. We just want to know which ones are allowed. So the error we've made here, so let me cross this out. So we know that's not the right approach. I've made a mistake in doing that. And the mistake is that the Z component of the dipole actually depends on this integration variable that I haven't bothered to write down yet. Remember, for the harmonic oscillator, the variable that the wave functions depend on is x, the bond displacement, the amount of stretch in this chemical bond, the covalent bond. So when x is positive, the bond has been stretched. When x is negative, the bond has been compressed. So I'm integrating over x. And in fact, the dipole moment of this molecule does, in fact, depend on x. We may like to think of HCl as just having some constant dipole moment. But in fact, there is some dependence on x. I can say it's got some constant dipole moment. And then there's some rate of change, some linear term. It either goes up or down as I stretch the bond. And in fact, it will go up. This dipole moment, if I stretch the bond and make it longer, then the bond is longer. This vector has gotten longer. The distance between the atoms has gotten longer. I also have to consider the size difference of the charges on the two ends of the molecule. But the dipole moment will get longer as I stretch the bond. So it depends on the value of the stretch of the bond. So the more correct way to approach this problem would be to say my integral looks like initial wave function. The dipole moment, the z component of the dipole moment looks like some mu naught, some default dipole moment when the bond is not stretched at all. And then some d mu dx, how quickly the dipole moment changes as I stretch the bond, multiplied by x. And then the ending state that I'm interested in. So that's what I actually want to integrate over x. And when I do that, let's go ahead and break this integral up into two pieces, one of which involves the mu naught term. So that's going to look like wave function mu naught second wave function. I've got psi n star mu naught psi n prime dx. The second piece is going to be from the second half of the term in parentheses. So that integral is going to look like psi n star d mu dx times x times the second wave function psi n prime integrated over x. Now, because of how I described the problem, there's some constant original dipole moment plus the change that reflects the dependent on x. This term is constant. This is just the default dipole moment when there's no stretch to the bond. So that I can pull out of the integral. And what's left will then be just psi n times psi n prime. And that integral will, in fact, be 0. So the first term is not important. Doesn't matter how big the dipole moment is, whether the dipole moment is large or small, this term is going to go away completely. The second term, what I can pull out of the integral, at least approximately in this case, is I certainly can't pull the x out of the integral. That's the variable I'm integrating over. But what I can, at least approximately, pull out of the integral is the rate of change of the dipole moment as I change x. If I stretch the bond a little bit and the dipole moment goes up a tenth of a dubai per angstrom, if I stretch it a little bit more, it'll go up by roughly the same amount. So if I pull this term out of the integral, I get this result. So now I have two-wave functions sandwiched around x. And I don't know for sure that this is going to be 0. So that's going to take a little more thought, and we'll postpone that until the next video. But remember, what we're interested in is finding out when this transition dipole moment is equal to 0. And that means the molecule cannot absorb light to make that transition. And when this integral is not equal to 0, and then the molecule can absorb light to make that transition. That'll tell us which of these transitions is allowed or forbidden. So if we look for ways to make this integral 0, this part has already gone away. If I look for ways to make this integral equal to 0, I can either do it by making the term out front equal to 0 or by making the integral itself equal to 0. So the integral is a little more complicated. That's what we'll postpone. But certainly, if that quantity is 0, then both terms are 0, and the transition dipole moment disappears, and the transition will be forbidden. So what that leads to is our gross selection rule, which I'll put here. I'll say d mu dx, the rate of change of a dipole, the amount by which the dipole changes as I change the bond length. That must be non-zero in order for a molecule to absorb light and change its vibrational state. If a molecule has d mu dx equal to 0, then both these terms will be 0, the transition dipole moment will be 0, and the transitions are forbidden. So for example, if I have some molecules like, I'll just list a few molecules, HCl, the one we've just been talking about. As we've illustrated, if I stretch that bond, the two charges get further apart. The dipole moment changes. So the dipole moment is changing as I change the bond length. I don't need to know the size of this number. I know that the dipole moment changes as I'm changing the bond length. So that molecule can absorb infrared light. We say that that molecule is IR active. In other words, it's active in the infrared spectrum. If I shine infrared light on this molecule, there are some frequencies that it will absorb. Likewise, other molecules with dipole moments, let's say carbon monoxide that has a relatively small dipole moment, but it has a dipole moment. That molecule is IR active. That dipole moment will change as I change the bond length. On the other hand, if I take non-polar molecules like H2, or N2, or O2, or Cl2, and so on, those molecules. So here's an H2 molecule. It has no dipole moment. The dipole moment is 0. But more importantly, when I stretch the bond, the dipole moment is still 0. The two Hs have the same charge. The dipole moment is not changing when I change the bond length. So because it's violating this gross selection rule, those molecules will not absorb infrared light. So these will not absorb infrared light. We say that those molecules are IR inactive, meaning that they are not active in the infrared spectrum. If I shine infrared light on these molecules, they won't absorb any of it in order to change their vibrational energy levels. So that leaves us with one question that tells us how to choose, how to know, which molecules can absorb light at all. We haven't yet made any progress on knowing which pairs of energy levels we can make transitions between. So to do that, we'll have to go back and look at this integral and find out which ends and end primes leave this integral 0 or non-zero. And that'll tell us the rest of the selection rules for the harmonic oscillator.