 So, if we have a diatomic molecule with several different energy levels, so here's a molecule with several different rigid rotor style energy levels that it can occupy, we've determined that the energy difference between those energy levels can be written with this formula, and the tilde on top means we're doing that in units of wave numbers, and that lets us predict what frequencies of light or what wave numbers of light will be absorbed by that molecule corresponding to the transitions between some state and the next state up. So the small peak that you can barely see right here would correspond to the zero to one, there's another peak corresponding to the one to two transition, that's this one, the two to three transition would be this one, and so on. But this raises a question, why are we limited to only jumping up one level at a time on this energy ladder? Why can't we jump directly from the L equals zero state all the way up to the L equals three state or to the L equals ten state or any state that we wish? And it turns out there's a good reason for that and we call those reasons selection rules. So to understand those selection rules, we need to know the following equation which tells us how strongly a molecule will absorb light in making a transition from one state to another state, maybe the L equals zero state up to another state or the way we'll talk about it as from one initial wave function to some final wave function. So the intensity with which that molecule in state psi will absorb the right frequency of light to make a transition up to the state psi prime, we can write that as the following somewhat complicated equation. There's a first term which depends on the difference in energy between the two states and the frequency of light that we feed it. And there's another term that looks a little bit like integrals we've seen before in quantum mechanics, but now sandwiched between those psi and psi prime wave functions. We have this new quantity mu sub z. So that is, we'll talk about that one in just a second. And I guess I'll say that there's also some constants out front, so rather than saying this is equal to, I'll say this is proportional to this quantity. So if we want to know whether a molecule absorbs light or doesn't absorb light, absorbs light strongly or absorbs like weakly or perhaps doesn't absorb light at all, we need to calculate this quantity, which is composed of two portions that we'll talk about separately. This first term, we'll talk about only in order to say this term tells us that if there's a particular delta E between two of these states, then this term guarantees that the frequency of light absorbed will be equal to or very close to that difference in energy. So normally, we know that delta E is equal to h nu. If we feed the molecule some light that has h nu equal to that difference in energy, then it will absorb light because this term will be strong. If I make a graph of sine squared of something divided by that thing squared as a function of x, the graph of that quantity is large when x equals 0 and oscillates and quickly dies and becomes small as x gets larger. So this term is large. The intensity of the absorption is large when x is equal to 0. For us, the quantity inside the sine squared and inside the denominator that gets squared, that's the difference between the energy of the transition and the energy of our photon. So when those things match exactly, the absorption is strong. If I deviate the frequency of light a little bit from what would be needed to induce that energy transition, the absorption is going to be quite weak. So in other words, if I want to induce this transition, I need to feed the molecule of light with a photon with that much energy, very much different than that energy and it won't absorb the photon or it'll have a very small chance of absorbing that photon. So that's really all we'll say about this first term. This second term, we'll go ahead and give this one a name because we'll be talking about it a fair amount. This is called the transition dipole moment and the reason for that name, transition becomes because we're talking about this transition between state psi and state psi prime. The dipole moment is because we're talking about we need to understand the dipole moment of the molecule that is absorbing the light. So this is the dipole moment of the molecule. Dipole moment would normally be a vector. Let's say if I have molecule like HCl that has a positive charge on one end and a negative charge on the other end, there's a dipole moment associated with that vector. Chemists tend to draw that vector from positive to negative. So the dipole moment of that vector has some direction. It also has some magnitude. This mu sub z and we would call that that vector mu. Mu sub z is the component of that vector in the direction that is the same as the direction of the oscillation of the electric field of the photon that we're shining on the molecule. So if I take a photon, shine it at this molecule and I'm going to ask does this molecule absorb light? Does it absorb this particular photon in order to make a transition from one state up to another state? It will if the frequencies match and if this condition is well satisfied also. So the dipole moment we're talking about here is not the vector of the dipole moment that I've drawn here but the component of the vector, if I break this vector down into a component that's in the same direction as the oscillation of the light and a component that is parallel to the direction of propagation of the light. In this 2D example then this piece would be the mu sub z that we're talking about. So it's the piece of the dipole moment that is in the same direction as the light. And so all we need to know for right now is in our spherical coordinate system mu sub z is equal to mu times the cosine of the angle theta. So that's how we're going to end up using this mu sub z in this integral. So when we want to look at this part of the calculation a little more carefully for a rigid rotor the transition dipole moment will have, if we want to talk about the transition between let's say quantum state described by L and M up to a different quantum state described by a different L and a different M that we'll call L prime and M prime. Then the transition dipole moment, the thing that looks like the initial state multiplied by the dipole moment in the z direction mu cosine theta multiplied by the final state. We need to integrate that over d theta and d phi because since we're talking about a rigid rotor a rotating molecule those are the coordinates in which our wave function is going to be expressed and don't forget when we integrate in polar coordinates we need the sin theta term inside the integral and then we're going to integrate over both of those variables. So that's what the transition dipole moment looks like for a rigid rotor molecule. We need to know what the initial and what the final wave functions are and then we've got the dipole moment term and integrating over theta and phi. So the question is when is that integral large and when is it small? There's one very easy answer to that question. There's one way to make that integral zero quite simply and that is by looking at this term. Mu, the mu that's in this z component of the dipole moment depends partially on what angle this molecule makes with the z axis but the mu without the z that's the magnitude of this dipole moment. So for a molecule like HCl that has a dipole moment that dipole moment is some particular value. Every molecule has a different dipole moment that's particular to that molecule but some molecules have no dipole moment. So for example a homonuclear diatomic molecule like H2 or Cl2 or N2 any homonuclear diatomic molecule has no dipole moment. They're non-polar molecules so for a non-polar molecule with mu equal to zero sticking a zero in this integral guarantees that the integral is going to come out to be equal to zero. So what that means is for those non-polar homonuclear diatomic molecules this whole transition dipole moment works out to be zero. This quantity i works out to be zero and what that tells us is light will not be absorbed. Regardless of what the orientation of the molecule is, regardless of what the frequency of light is, regardless of which transition we're trying to make, this quantity is zero so that's called a forbidden transition. Light will not be absorbed in order to cause that transition. Again doesn't matter which transition I'm trying to cause I cannot shine microwave light on a chlorine molecule and cause it to change its rotational motion. So that's an example of what we call the gross selection rule. For rotational motion we need the dipole moment of the molecule to be non-zero in order for the molecule to absorb light. So we can get molecules like HCl to absorb light at least for some transitions because they have a non-zero dipole moment but for non-polar molecules because they violate the gross selection rule then those transitions are all forbidden and they do not absorb light in the microwave. We can't cause rotation of a non-polar molecule using electromagnetic radiation. That doesn't mean we can't cause these molecules to rotate. Hydrogen, nitrogen, chlorine molecules do rotate but the rotation has to be caused by some other source such as collision with another molecule rather than by absorbing light. So this raises the question of are there other ways to make this integral zero and in fact there are, so that's the next thing we'll talk about is a slightly more complicated question of what do these wave functions have to be in order to make this integral equal to zero or not equal to zero and that will be not the gross selection rule but another flavor of selection rule.