Quantum Chemistry 6.11 - Orthonormality of Spherical Harmonics

[@TheRogueThunder]

I don't understand where that equation came from...

[@rickyzeitler7169]

Wouldn't the new bounds of the integral be from pi to zero, and wouldn't having it be from zero to pi require a negative sign in front of the integral?

[@rickyzeitler7169]

Oh, its because x = cos(theta), dx = - sin(theta)d(theta), so those negatives would cancel out.

[@TMPChem]

Indeed. Flip the bounds, flip the sign. #CalculusMagic

[@praveenmishra5884]

Is there also spectroscopy lecture of polyatomic molecules particularly of asymmetric top (asymmetric rotor)? Please suggest

[@TMPChem]

Hi Praveen. Unfortunately I do not have a video on the rotational spectra of anything more complicated than a diatomic molecule, as that is typically more advanced than what is taught in an undergraduate spectroscopy course. The best I have is a video in the computational chemistry course on computing the rotational constants of a general polyatomic molecule using Python, but that video does not discuss any further details about the rotational spectrum.

[@nkyu3035]

hope you understand I tried to ask fewer question T_T Why does the first green integral results in different value after substitution? it's multiplied by 1/(2J+1) ?! you defined Capital Phi to have a coefficient of sqrt(1/2pi). shouldn't there be 1/2pi in front of orange term on the right? and right below it,

[@TMPChem]

Eigenfunctions of Hermitian operators are required to be orthogonal (if non-degenerate), but we should be able to verify that through calculation as well. Rigid rotor states with the same J are degenerate, so it's important to show that choosing a different value of m leads to orthogonality as well. I believe I've defined Phi(phi) to have that normalization constant in the wavefunctions video, though the functions presented here are pre-normalization, and demonstrate where those normalization constants come from. As for the factor of (2J+1), I'm not sure. I remember having to do a lot of piecing things together from various sources, and I can't find that one at the moment, and I don't know it off hand. I can say that Legendre polynomials are orthogonal over the range -1 to +1, but that doesn't imply anything about normalization.

[@nkyu3035]

Really thank you for the comment. the vastness of this field scares me -_-;;

[@TMPChem]

Most technical sub-fields start feeling quite vast at the advanced undergraduate / early graduate level. That's when most people reach a level of education where they start becoming aware of how much there is to know that they don't know. That's why many people finish advanced studies feeling like they know less than when they began, having spent the better part of a decade studying a topic only to realize how much more they have left to learn, and that there is that much to learn in almost any technical topic imaginable.

[@TheYetiMr]

Hi! This may sound like a stupid question, but I thought d(cos(theta))/d(theta) = -sin(theta) and so dx = -sin(theta)d(theta). Am I missing something?

[@TMPChem]

I believe you're correct about the derivative. There may or may not need to be a factor of -1 somewhere to fix this, but luckily for the purposes of this video it doesn't affect the final result. The goal here is to show that the overlap integral of two rigid rotor wavefunctions is zero when the quantum numbers differ, and non-zero when they are the same. Being off by a factor of -1 will change the sign of the result, but will not change whether it is zero or non-zero. In some refrences I see a factor of -1 to some power preceding the wavefunctions. Perhaps this is the problem that that prefactor is included to solve.

[@TheYetiMr]

Understood. Thanks for the quick reply! Really appreciate the videos.

[@AndreaLascala-x1s]

@@TMPChem When you write [-1,1] -> [pi, 0], by putting in crescent order the limits of integration, you obtain another minus sign that cancels out with the sign that comes from the derivative.