[lukko]

This is really great. I always saw those harmonic shapes as electron orbitals, I had no idea they could be used in lighting too - so cool.

It made me wonder - why do the electron orbitals take those shapes in say a hydrogen atom? Is there a constraint on the electron and proton together that make it fit only to spherical harmonic functions?

[rsfern]

The reason is that electrons (like all quantum mechanical objects) are wavelike. In an isolated hydrogen atom, the electron is in a spherically symmetric environment, so the solutions to the wave equation have to be spherical standing waves, which are the spherical harmonics. The wave frequencies have to be integer divisions of 2pi or else they would destructively interfere. (Technically each solution is a product of a spherical harmonic function and a radial function that describes how fast the electron wave decays vs distance from the nucleus)

What’s interesting is if the environment is not spherically symmetric (consider an electron in a molecule) the solutions to the wave equation (the electronic wave functions) are no longer spherical harmonics, even though we like to approximate them with combinations of spherical harmonic basis functions centered on each nucleus. It’s kind of like standing waves on a circular drum head (hydrogen atom) vs standing waves on an irregular shaped drum head

Of course the nucleus also has a wave nature and in reality this interacts with the electrons, but in chemistry and materials we mostly ignore this and approximate the nucleus like a static point charge from the elctrons perspective because the electrons are so much lighter and faster

[lukko]

Ah amazing - thank you for the response! I have a couple of related questions - is it that the non 2 pi frequencies exist, but they destructively interfere so we can't see them? My understanding is that the radial function for the electron is zero at the nucleus - there is no possibility of it being found there - but why is that the case?

[rsfern]

Admittedly my understanding of QM is a bit vibey but I’ll try to answer

In an atom, angular wavefunctions with wavelengths non-integer divisions of 2pi can’t exist because of the boundary conditions on the wave equation. A free electron can have any wavelength, but once you put it in a box (confine it to the potential around a proton in a Hydrogen atom) the non-integer wavelengths aren’t allowed

I think it’s instructive to think about what the wavefunction represents. It’s square is the electron probability density (technically the wavefunction is complex valued so it’s the wavefunction times it’s complex conjugate). If you have a non-integer multiple wavelength then the wavefunction goes out of phase with its complex conjugate after one period, and if you integrate over the angular domain the electron probability has to be zero everywhere.

This also answers your second question. The radial solution to the wave equation for hydrogen gives you the Laguerre polynomials. They don’t all go to zero at the nucleus though, actually the first one has a maximum at zero because it scales like exp(-r) (See fig 4.10.2 on chem.libretexts linked below). But when you do a volume integral to calculate the electron probability, the probability near the nucleus is low because the integration volume is small even though the wavefunction is large

https://en.wikipedia.org/wiki/Laguerre_polynomials

https://chem.libretexts.org/Courses/University_of_California...

[crispyambulance]

Spherical harmonics are basically a fourier series. They're a complete orthonormal set of basis functions for functions for the unit sphere. Whereas the fourier series from calc 101 is a complete orthonormal set of basis functions on the unit interval (eg [0,1]).

In other words you can express any reasonable function on the unit sphere as a series of spherical harmonic terms. That makes them ideal for working with differential equations (eg schrodinger's equation for the hydrogen atom, or, emission from an arbitrary light source).

[groundzeros2015]

And the number of terms you need to get a good approximation is related to the frequency. Low frequency signals like lighting work well.

[lukko]

this is all so interesting.. Are there any particular functions / parameters that are typically used, that say replicates 3 point light setups?

I guess at a certain point the number of terms becomes so large that it makes sense to just use a cube map?

[groundzeros2015]

In the era im familiar with (ps3, 360) everyone used the first 9 coefficients. You can read the original Ramamoorthi paper for better theory applied to lighting.

But yes it’s an approximation. If you have a ton of terms it looks like a bitmap like you said.