 So, we're interested in this transition dipole moment when under what conditions this is equal to zero and a transition is forbidden and under what conditions it's not equal to zero and the transition is allowed and in particular what values of l and m make this forbidden or allowed. So, to explore that a little further, let's recall for the rigid rotor, specifically the form of these wave functions for the rigid rotor, the wave function for quantum numbers l and m, angular momentum quantum number l, magnetic quantum number m. There's a normalization constant, there's a piece that looks like a Legendre polynomial in cosine theta and then there's the exponential term that has this e to the i m phi with the quantum number m. So, if we're thinking about how to plug rigid rotor wave functions into this expression in the general case, that's the form we'll use. The lm wave function with the complex conjugate, if I write down what this looks like for the complex conjugated form, the constant might need to be complex conjugated, the Legendre polynomials are not imaginary, they don't have any imaginary portion so I don't have to do anything there and for this e to the i m phi, the complex conjugate just turns the i into a negative i and then if I write down what that looks like for the l prime m prime piece, it's just going to be a different normalization constant, a different Legendre polynomial and then e to the i m prime phi that doesn't get a negative sign because that one doesn't get complex conjugated. If I insert those into the transition dipole moment, what we have is, and I'll go ahead and simplify a little bit as we do this, pulling the constants outside of the integral wave function number one, wave function number two, both have constants, the dipole moment is constant so the constants out front are two normalization constants and a dipole moment. The integral, the theta dependent portions of the integral, I've got a plm from wave function lm and I've got a pl prime m prime from the second wave function and then I've also got some cosine theta's one here and a sine theta from here and then, so that's my theta integral, the phi integral, I have a d phi here and then the phi dependence of these wave functions is this e to the minus i m phi, e to the plus i m prime phi multiplied by d phi and integrated over phi. So that's the integral we need to evaluate to determine whether the transitions forbidden are allowed, in particular looking for anything that could cause either of these two integrals to go to zero. So we'll go ahead and do that and we'll start with the integral, the phi integral that involves this magnetic quantum number because that one's a little bit easier and we'll consider two separate cases. Case number one, let's consider the possibility that the final quantum number is equal to the initial quantum number. So the quantum number, the magnetic quantum number is not changing as the transition happens so we could also say that as delta m is equal to zero. When we do that, if we evaluate this particular integral, I'll write that integral as e to the i m prime minus m phi, just combining these two exponentials into one, so that's the integral I want to perform, m prime minus m, that's the same thing as this delta m and under this case that we're considering when delta m is equal to zero, this integral e to the zero phi, that's just one. So because we're considering the case where the quantum number is not changing, this exponential is one and I'm just integrating d phi from zero to two pi which evaluates to two pi, but more importantly it evaluates to something that's not zero, so that transition is allowed. If the quantum number m is not changing, this phi contribution to the transition dipole moment is non-zero and that doesn't kill the transition dipole moment, that transition is allowed. On the other hand, if we consider the case where the final magnetic quantum number is not the same as the initial quantum number, so this delta m is something non-zero, maybe it's increasing by one or decreasing by one, whatever it is doing it's changing, then this integral, integral of e to the i m prime minus m is the thing we're calling delta m, e to the i delta m phi integrated over phi from zero to two pi, that's just an exponential integral, it looks a little scary because it's got imaginary numbers up in the exponent, but the integral of an exponential is just an exponential with a one over i and delta m in front so that if I were to take the derivative of this function, that i and delta m will cancel this i and delta m and leave me with just the exponential that I was taking the integral of. So that is the integral, I want to evaluate that between zero and two pi, so I'm going to get one over i delta m, e to the i times delta m times two pi, so I'll write that as e to the two pi times i times delta m, and the second term I've got e to the i delta m times zero, so e to the zero is just going to be one. But this is a fairly convenient result, one of the few things we need to know about imaginary numbers to deal with them in these rigid rotor wave functions is the fact that we're going to use in this next step is that e to the two pi times i is equal to one, as is e to the four pi times i, as is e to the six pi times i, e to any integer, any even multiples of pi times i up in the exponent, that is equal to one, that's just a feature of imaginary constant i, so e to the two pi i maybe times one or negative one or two or negative two, some change in the quantum number, magnetic quantum number, this whole first term is equal to one, so what we have is one minus one inside the parentheses and this whole integral comes out to be equal to zero. So it turns out that anytime the magnetic quantum number does change, if there is a change in the magnetic quantum number, the pi contribution to the transition dipole moment will work out to be zero, that will make the entire transition dipole moment equal to zero and that will be a forbidden transition. So what that results in is the general statement that if we want to induce a transition between some initial state and some final state using electromagnetic radiation, using light, then the requirement is that the transition we want to initiate cannot change the magnetic quantum number, the delta m must be zero in order for that transition to be allowed. So that's the first of two selection rules that we can determine for rigid rotors, that's the one that determines how the value of m should behave. We can take a look at this theta integral and determine how the value of l should behave if we want that transition to be allowed and that's what we'll tackle next.