Talk:Table of spherical harmonics

Contraction of +-m harmonics
I just noticed that anonymous 193.175.8.58 made some edits in the middle of November, contracting the formulae of spherical harmonics with $$l=2$$. While this notation is indeed more concise, I have some objections. Firstly, the choice of only $$l=2$$ for the edit seems arbitrary. Why not fix the other ones as well? Secondly, I think the expanded version would be much more useful in the case of this specific article. As the article is a table, it doesn't matter that 2 (or $$l$$) more lines are used for these functions. However, when users access the page and want to use/check these functions, the expanded notation (I mean having separate $$+|m|$$ and $$-|m|$$ harmonics) is much more transparent, and provides much less room for errors. If it was up to me, I would revert to a preceding version, but others might argue to change all the other spherical harmonics instead. Either way, I'm hoping for some input from others. --Loudandras (talk) 20:36, 3 January 2014 (UTC)
 * I noticed that set of edits back then but was too busy to think about it. I agree that all sections should be consistent on this point. I have a small preference for reverting the edits as I agree that would be more transparent. -- Bduke   (Discussion)  20:54, 3 January 2014 (UTC)
 * Thanks for the feedback! I decided to revert the edits by IP. --Loudandras (talk) 16:20, 8 January 2014 (UTC)

Direction cosines
I see the listing 'drys up' at l=5 (going from x,y,z to theta, phi. Philip Cotterell did produce the direction cosine (x,y,z) equations upto l=6. They are at  etc.,but would need some converting to match your 'style'. (They were republished at 'AMBISONICS SYMPOSIUM 2009 JUNE 25–27, GRAZ' "Symmetries of Spherical Harmonics: applications to ambisonics" Michael Chapman (pages 2 and 4), if you need a more formal source. Various copies of the PDF are on the Web.) Hope this helps...? — Preceding unsigned comment added by 92.157.70.113 (talk) 04:44, 10 January 2014 (UTC)

Real harmonics, notation and order
Regarding recent changes to the section about real spherical harmonics: I think that the orbital notation was really helpful, I don't really see why this was removed. There is no actual meaning of the $$m$$ quantum number of real harmonics, I mean there can be no a priori distinction between real harmonics with $$+|m|$$ and $$-|m|$$. Furthermore, the atomic orbitals are actually used to denote these functions, for example in various chapters of solid state physics. Is there a particular reason to remove the orbital notation? And on a similar note: was there any reason to switch the two real harmonics with $$l=1, m=\pm 1$$? It shouldn't make any difference, choice of indexing is arbitrary. I'm just arguing that the orbital notation should also be kept.

OK, I reread the article and the one on spherical harmonics. The new notation here seems to be in accordance with that article, and this also gives a reason to the change in the harmonics for $$l=1$$. Maybe some more extensive edit comments would have been helpful with the latter. Still, I believe that the orbital notation should be kept. And perhaps the functions should be checked now to see whether the notation is completely consistent with the spherical harmonics article. -- Loudandras (talk) 19:18, 17 January 2014 (UTC)

I came here to make essentially the same comments. I think the atomic orbital notation should be added back, so both notations are there. I have this on my watchlist because, as a computational chemist, I have used them. I suspect most people who will use this page, at least for the section on the real spherical harmonics, will be chemists. -- Bduke   (Discussion)  21:21, 17 January 2014 (UTC)

Sure, it can be restored.. but I think the normal indexes should still have the priority, since real expansions often go far beyond the orbital notation. So what I suggest is that a separate section be added on what specific names have been given to the Y_{lm} in the physics and chemistry communities. This can be either a subsection for each l value, below the equations for the Y_{lm}, or a wholly separate section by itself in e.g. the Spherical harmonics page. Susilehtola (talk) 08:00, 18 January 2014 (UTC)

I put in the term symbols as well in the table. Is this OK for you guys? To reiterate, I think the normal indexes should appear first since they're used everywhere for performing spherical expansions, while atomic orbital (AO) symbols have rather limited use outside quantum chemistry. (I'm actually a quantum chemist myself and have never used the AO symbols since they are cumbersome in computer implementations.) Susilehtola (talk) 09:35, 18 January 2014 (UTC)
 * Yes, I am OK with that. I was going to suggest it. Now, could anyone add the real spherical harmonics for l=5 and l=6 as these are used in quantum chemistry quite a lot these days? -- Bduke   (Discussion)  10:01, 18 January 2014 (UTC)


 * I don't see much of a point in that; it's by far easier to implement the equations for the cartesian form on the computer for general angular momentum than to hand-code implementations for s, p, d, f, g and so on. Of course, it wouldn't hurt to have them. Susilehtola (talk) 19:01, 22 January 2014 (UTC)


 * Thanks a lot, this is what I had in mind as well. Someone suggested some work on l=5 and 6 real harmonics in the "Direction cosines" section of the talk page, and I found a corresponding pdf here. But citing those would involve checking the prefactors, converting the notation, adding a factor of 1/r^l, and linking each harmonic to the corresponding complex ones to keep the syntax of l=1...4. So... maybe some other source?:) --Loudandras (talk) 10:38, 18 January 2014 (UTC)

Could someone please provide a reference for how the atomic orbital names for the real orbitals are generated? I can't tell what the rules are. For example, for $$Y_{3, -3} = f_{y(3x^2-y^2)}$$ the numerator of the cartesian expansion is $$(3x^2-y^2) y$$, but for $$Y_{4, -3}=g_{zy^3}$$ the coeffecient of the numerator of the cartesian expansion is $$(3x^2-y^2)yz$$. Why aren't the subscripts on the atomic orbital notation names more similar? Twistar48 (talk) 16:26, 18 February 2022 (UTC)

I've been searching for a reference for the $$g$$ orbitals but I can't find it. Does the Chisholm reference tabulate them? I haven't been able to get a hold of it yet. If not we need to remove those labels. The only place I have seen the $$g$$ orbitals labeled is https://winter.group.shef.ac.uk/orbitron/, and there is no explanation there-in of how the abbreviated polynomials are derived from the full polynomials. None of the references therein address higher order polynomials either.

It's obvious the "atomic orbital" nomenclature arises from an abbreviation of the $$x, y, z$$ cartesian expansion of the corresponding real spherical harmonic. But nowhere is it explained how the polynomial is abbreviated. It's clear that $$z$$ part just comes from taking the highest order of $$z$$ in the $$z, r$$ polynomial. That is fine. For $$Y_{\ell m}$$ we will have $$z^{\ell - m}$$. The $x, y$ part of the polynomial is given by either $$\text{Re}((x+iy)^m)$$ or $$\text{Im}((x=iy)^m)$$ depending on the sign of $$m$$. What is not explained in any reference is how this $$x, y$$ polynomial is abbreviated to come up with the atomic orbital nomenclature subscript. In fact, for $$f$$-orbitals and below the common nomenclature does not abbreviate the polynomials. This means we end up with the dirty looking $$Y_{3, +3} = f_{x(x^2-3y^2)}$$ but at least it's unambiguous where it comes from. It isn't until the $$g$$ orbitals that we see the first abbreviation: $$Y_{4, +3}\propto xz(x^2-3y^2) = g_{zx^3}$$. At this point, when the $$x, y$$ polynomial is being abbreviated the nomenclature fails to provide any useful goemetric information about the orbital. Also, anyone tabulating orbitals this high has to know about the usual $$\ell m$$ spherical harmonic nomenclature. I think it is for this reason the atomic orbital nomenclature stops at $$f$$. The strange abbreviation on the $$g_{zx^3}$$ also breaks the previously established rule that orbitals with the same $$m$$ but different $$\ell$$ have the same $$x, y$$ part for their atomic orbital name. I've outlined this in more detail in a stack exchange question I asked and answered myself here https://chemistry.stackexchange.com/questions/163148/how-are-names-for-atomic-orbitals-with-high-l-generated. There I give a general closed form formula for all spherical harmonics which is easily derived from formulas appearing on a few Wikipedia pages.

For all of these reasons (1) I'm going to remove the $$g$$-orbital atomic orbital nomenclature names until someone tells me it is tabulated in the Chisholm reference or provides another reference. (2) I'm going to add 2 references for the atomic orbital names. (3) I also think we should have a description somewhere of how the atomic orbital nomenclature is derived. Unfortunately, again, this is not explained anywhere in the literature. If we stick to $$f$$ orbitals and below I think we could just say it comes from the $$x, y, z$$ expansion of the spherical harmonic and then dropping any terms then $$z, r$$ that aren't the highest exponent in $$z$$. One big question I have is where this explanation should go. Should it go on this page? The spherical harmonics page? Or one of at least 3 pages on atomic orbitals? I don't think it should go on the Spherical Harmonics page because that page doesn't really make reference to atomic orbital stuff, and I don't think it should be cluttered with that. I think an explanation should go on the atomic orbitals page and be linked to on this page. I may work on that. (4) Finally, I would be happy to work on adding the expansions for the real spherical harmonics up to 10, but I would be using the formula I derived in the link above (will post it here) along with mathematica or something for checking. If this is legit, I'll do that. If we require a referenced tabulation, I won't do that, but it seems a shame to leave them off when they're easily calculated.

Finally, I'm going to take this moment to advertise why I've become so interested in this. I'm working on a gallery of atomic orbitals page, inspired by this one, that can be found here: https://en.wikipedia.org/wiki/Draft:Gallery_of_Atomic_Orbitals. it was recently brought up on that page that I should label my orbitals using the atomic orbital nomenclature. My orbitals on that page go up to $$h$$ so I was trying to find tabulations, but the only place I found any tabulations above $$f$$ was here and The Orbitron. I took the approach on that page of giving orbitals $$f$$ and below the atomic orbital names, but foregoing them above. Twistar48 (talk) 16:33, 20 February 2022 (UTC)

Recent edit
Please take a look at the recent history. I reverted an unexplained change, but it seems as if it might indeed be a small error. The discuss is at User talk:Fegp59. As I say there, I am on vacation and do not have access to my usual sources. In fact even when I return, many will be in a stored container!. -- Bduke   (Discussion)  10:59, 3 November 2015 (UTC)

ℓ instead of l
I think we should use "ℓ" and not "l". It's much easier to read, and is the standard, and also is used in many Wikipedia articles, like Azimuthal quantum number. What do yall think. I can implement the changes if yall think it makes sense. -- Blue.painting (talk) 16:49, 13 February 2019 (UTC)
 * Even more importantly, it's also used by Spherical harmonics so the notation would also be clear. I think you should go for it, assuming that using $$\ell$$ doesn't mess with the table of contents. But I'd use a math-typeset \ell rather than a unicode character. -- Loudandras (talk) 17:04, 13 February 2019 (UTC)
 * Ok, great, I just changed stuff, replacing all "l" with "$$\ell$$". TOC still works fine. Cheers. -- Blue.painting (talk) 10:08, 16 February 2019 (UTC)

Added visualizations
I added extensive tables of visualizations for the complex and real spherical harmonics up to l=10. For each type I include three types of (redundant) visualizations. In one case I show the magnitude and phase of the spherical harmonic on a 2D plot with polar (theta) and azimuthal (phi) angles on the axes; magnitude is represented by saturation, phase is represented by hue. In another case I essentially wrap the previous plot onto a sphere in 3D by using a polar plot. Magnitude is still represented by saturation and phase is represented by hue. In the final case, I again use a polar plot, but now magnitude is represented by the radius of the polar plot (and saturation is fixed to max everywhere).

I will post a link to a github page with code that can be used to generate these plots. These plots were generated using python scipy special functions for the spherical harmonics and pyvista for 3D polar plots. I believe they are in accordance with the conventions on this page and the Spherical harmonics page, but one can only be so sure with how many possible conventions there are.

I could hear arguments that there are too many plots included. The first argument could be that the three separate representations are unnecessary. However, I find it helpful to aggregate these three visualization techniques here for the following reasons. The 2D theta/phi maps are useful because they emphasize that spherical harmonics are simply 2D functions of these two angles. I think an analogy with Fourier series is also evident from looking at the periodic or checkerboard structure that arises on these plots. The constant-radius polar plot is useful because it demonstrates how this function looks on a sphere, and the final polar plot is useful because it makes the magnitude even more evident than on the previous plot. I find it useful to include both polar plots because, the reader will come across both version in the literature and here on Wikipedia on the Spherical harmonics page. Here it is clear that they are 2 representations of the same information, and the reader can compare and contrast.

The radius-as-magnitude version is useful when working with atomic wavefunctions, but, when visualized with this way, a beginner may confuse spherical harmonics with the atomic orbitals themselves. The constant radius plots help make it clear, by point of contrast, that spherical harmonics are not atomic orbitals, just closely related.

Finally, the second argument I could hear against these tables is that it is too much to include up to l=10. I only chose l=10 because explicit expressions have been included for the complex spherical harmonics up to this value. I think it's important to include the simple ones, say up to l=4. But it is nice to include some more complicated ones, up to say, l=6 or 8 so that the reader can test their pattern recognition (i.e. their ability to extract l and m from a given pattern, or predict the shape of a given harmonic given l and m) up to high values of (l, m). Anyways, if some think l=10 is too much I'd hear arguments and could generate plots with lower l.

Finally, I think it may be appropriate to include a section explaining the shape of the spherical harmonics as a function of l and m. In particular that the number of polar nodes (excluding the node at $$\theta = 0, \pi$$) is given by l-m and the number of azimuthal windings is given by m (for complex harmonics, for real harmonics the number of azimuthal nodes is given by 2m). I may add this later.

Twistar48 (talk) 07:28, 5 November 2021 (UTC)


 * Here is the repository that was used to generate these visualizations. https://github.com/jagerber48/hydrogen_visualization. Twistar48 (talk) 23:07, 7 November 2021 (UTC)