So the spherical harmonic functions are also the shape of the various electron orbitals that you see in a chemistry textbook.
Minor nitpick: Chemistry textbooks usually use the “cubic harmonics” instead of the “spherical harmonics”. They are real-valued linear combinations of the standard spherical harmonics, with the additional benefit that the basis set respects the Cartesian symmetries.
For example, the “p_z orbital” is a l=1 spherical harmonic and a cubic harmonic. But the cubic harmonics then add p_x and p_y orbitals as basis functions, whereas the spherical harmonics choose the chiral "p_x ± ip_y orbitals” as its basis instead.
Oh thanks, I didn't know that (I studied physics and only know the chemistry part superficially). But I did know, and wondered about, the fact that the spherical harmonics have an e^imϕ factor yet there's nothing about the explanation that involves complex numbers per se (besides that writing out sine/cosine series is way more tedious than exponentials). Makes sense that they just get factored.
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Excellent explanation.
Minor nitpick: Chemistry textbooks usually use the “cubic harmonics” instead of the “spherical harmonics”. They are real-valued linear combinations of the standard spherical harmonics, with the additional benefit that the basis set respects the Cartesian symmetries.
For example, the “p_z orbital” is a l=1 spherical harmonic and a cubic harmonic. But the cubic harmonics then add p_x and p_y orbitals as basis functions, whereas the spherical harmonics choose the chiral "p_x ± ip_y orbitals” as its basis instead.
Oh thanks, I didn't know that (I studied physics and only know the chemistry part superficially). But I did know, and wondered about, the fact that the spherical harmonics have an e^imϕ factor yet there's nothing about the explanation that involves complex numbers per se (besides that writing out sine/cosine series is way more tedious than exponentials). Makes sense that they just get factored.