Oh Hund

Readers who are chemists or in related fields:

I have a question where I appear to disagree with Eldest’s chemistry honors teacher.

It’s about the (ground state) electronic configurations of atoms. It’s not a problem as to which orbitals get filled first; it’s slightly more subtle. Namely, once you start filling a degenerate set of orbitals (the set of orbitals with the same n and l, having the same energy), our good buddy Hund helpfully pointed out (the maximum multiplicity rule) that first you put one electron with spin up in each orbital; only after all the orbitals with a given n,l have one spin up electron do you start putting  the second electron (now with spin down) in each orbital. For a given l, there are 2l+1 orbitals, with magnetic quantum numbers ml going from -l to l, in unit increments.

Now the question is: when you start filling the orbitals with a given l and different ml, which ml’s get filled first?

I say from highest to lowest (it has to do with maximizing the orbital angular momentum, similar to what is done with spin). Eldest’s teacher says from lowest to highest. Eldest’s book and most online resources don’t discuss this, but those that do confirm what I remember. Chemists, what say you?

Eldest came the other day with an assignment where the question was what the n, l, ml, and ms (spin up or down) of the last electron added for a given atom is. They did an example in class where they started filling up a three degenerate p orbitals (l=1) from ml=-1 and up. I said I thought it was wrong, checked online, and though he might have copied it wrong in class.  I told him to go explicitly ask his teacher today; he did, and yes, she says from lowest to highest. I am pretty sure I am right and she is wrong, but I don’t have any undergrad chem textbooks lying around to check and my trust in Wikipedia is vast but not unlimited.

Another question is what to do if the teacher is wrong. It’s easy enough to tell my kid “I am pretty sure she is wrong, but keep your head down and be prepared to learn differently in college.” The thing is that the order of filling ml’s is a relatively minor point at this level (he’s a high school sophomore); virtually nothing in his upcoming few years depends on knowing the order of filling orbitals in the right ml sequence one way or another. I also don’t want to be a douche professor parent, throwing my academic weight around and appearing to bully a high school chemistry teacher; after all, she can take it out on my kid. But if bugs me that hordes of students might learn something wrong. I will likely just let it go, though.


  1. I have 20+ hrs of chemistry and I don’t recall any of this!

    If you get confirmation that you are right, There are 2 approaches I’d take. If your son is also interested is researching the correct answer, I’d encourage him to bring it up in class or privately with the teacher. Have him bring documentation & coach him on how to politely bring this to her attention (my mom was helping me and we weren’t sure so we looked it up…). If he isn’t interested, I’d do the same myself probably via a short friendly note (I was helping x with his homework and ran across this, which is different from what I was taught. According to this blurb from textbook (attached) this is what it should be which is different from the assignment x brought home….end with a nice sentence about how x enjoys/is learning a lot in her class). Then the ball is in her court & hopefully if she cares, she will correct her mistake now and in the future. If not you’ve done your duty, she hasn’t lost face, no big deal.

    Curious to hear what others think.

  2. So the confusion here is one of notation: when there is no magnetic field, all of the m_l levels are degenerate, so there’s no meaning to the question “which fills up first” – in reality the first electron will be in a superposition of all three orbitals. The article you cite (“Lecture 27”) is confusing on this point – it is USEFUL to say that you are adding electrons to the highest m_l first because it helps you figure out the L value for that electron configuration, but this is purely bookkeeping. This point is not stressed enough in most texts and online sources.

    Once you add a magnetic field, then the m_l orbitals separate in energy, with the lowest m_l having the lowest energy.

  3. Josh is right. You should really think about maximising S^2 and L^2 in a rotationally invariant system see eg section 2 of Antoine Georges, Luca de’ Medici, and Jernej Mravlje, Annu. Rev. Condens. Matter Phys. 2013. 4:137–78

  4. Josh, this isn’t about the Zeeman effect (splitting in a uniform static magnetic field). it’s about the fine atomic structure. This isn’t discussed much in low-level chemisty, but here goes. *Empty* orbitals with a given l but different ml are degenerate. However, once you start putting electrons in, depending on their total spin and orbital angular momentum, the degeneracy breaks because of spin-orbit coupling. Atomic configurations with a certain number of electrons in, say, p orbitals, but with those electrons being distributed differently over spin and ml, will give different *total* energies. That’s what Hund’s rules capture, that the lowest (ground) state of a multielectron atom will have the maximal multiplicity of spin (first fill all with spin up, only then with spin down) and maximal multiplicity of orbital angular momentum (fill with high ml first). We teach it to kids as the orbitals being degenerate and then we put multiple electrons in them while obeying these strange filling rules, but in reality the total energy for different configurations is slightly different and the rules we teach are for the ground state (the other configurations aren’t anathema, they are simply excited states, with slightly higher energies). See Hund’s rules .

  5. xykademiqz I found a good reference explaining that the configuration with highest angular momentum is the lowest energy in McQuarrie’s Physical Chemistry (ends on pg 301 in my addition with a summary of Hund’s rules). However this is not something normally talked about explicitly in college level Gen. Chem. classes (often avoided by not asking for the exact last electron placement but instead for possible placements).
    I think your response depends both on Eldest’s personality and that of the teacher. Will Eldest feel comfortable/gain confidence from talking to the teacher about this and is the teacher going to be open to taking information from a student or a parent? Honestly the teacher was probably surprised by Eldest’s question and just went with the first answer they thought of with out really thinking. If I was the teacher I would love to learn the reasoning behind the the correct method (I thought your explanation here in the comments was very clear without being too technical).
    As a student of science parents, I was often sent by my mother to correct teachers and some responded well and others not so much. But my mother always let me try to talk to the teacher first on my own before stepping in to back me up if they balked at being corrected by a student. I want to know what happens!

  6. Hi XY,
    Everything you are saying is correct except the “first fill with all spin up/high ml first” parentheticals. Think about the multiplicities of the states: Suppose we have a p2 atom: the term symbols are 1D, 3P, and 1S. By Hund’s rule, the 3P is in fact lowest energy because of the highest spin multiplicity. But once we form these states, it doesn’t make sense to say “electron 1 is in orbital p1 and electron 2 is in orbital p0. The multiplicity of the 3P state is 9, so there are 9 equivalent “microstates” that contribute to it: [alpha(p-1),alpha(p0)],[alpha(p0),alpha(p1)],etc. Simply the fact that the state is 9-fold degenerate means that you can’t say “the first two electrons specifically go in to the two lowest m_l orbitals”.

    Another point is that in the absence of the frame of reference and breaking of degeneracy given by an external field (for example), there is no way of distinguishing between a p-1 and p1 m_l state . This is the same as the m_s states – if you have an S orbital (so we’re not worrying about angular momentum coupling), it makes no difference if you put an electron in spin-up or spin-down. We tell students to put the first electron in spin-up to make things seem easier, but if you put the electron in spin-down it would have the same energy (and really it’s both until there’s a reason for it to be one or the other)

  7. I think what you outlined below is the right approach–its tough being a teacher and if its not a fundamental problem….

    “The thing is that the order of filling ml’s is a relatively minor point at this level (he’s a high school sophomore); virtually nothing in his upcoming few years depends on knowing the order of filling orbitals in the right ml sequence one way or another. I also don’t want to be a douche professor parent, throwing my academic weight around and appearing to bully a high school chemistry teacher; after all, she can take it out on my kid. “

  8. Josh, there are several Hund’s rules, speaking of spin magnitude (maximized), orbital angular magnitude momentum (maximized), and total angular momentum (|L-S| or L+S depending whether the level with given l is less or more than half full). It seems to me that above you are are only talking about the rule for spin, but there is more and technically the degeneracy does get broken further when you take spin-orbit interaction into account and the ground state has a very specific S, L, J (and the degeneracy of 2J+1, as revealed e.g. in Stern-Gerlach experiments).

    But yes, I agree with your general sentiment that “up” and “down” (also positive or negative ml) is generally meaningless unless there is something to break the isotropy, such as external magnetic field or spin-orbit interaction. If we are talking about the energy levels without spin-orbit coupling (which, let’s face it, a kid in sophomore year of high school probably is) then all orbitals with a given l are truly degenerate and it’s silly to ask which get filled first. However… If we are at that level of approximation, then we shouldn’t be insisting on filling each orbital with one electron of a given spin orientation first until half full and only then adding a second, which everyone insist on; we should say that as long as you don’t have electrons with the same spin in the same orbital and thus don’t aggravate the ghost of Wolfgang Pauli, you are good to go and all such configurations are allowed and all have the same energy.

    But… We *do* teach the kids to mind Hund’s first rule, sort of (maximize total spin magnitude to lower energy, which we teach as put all electrons with one orientation of spin first etc.) It seems we pretend that the equivalent rule for orbital angular momentum doesn’t hold.

  9. By the way, this nerdy discussion made me very very happy (thanks Josh, Lisa, et al.!) and it further elucidated what a really really fine point this ml-filling business is. I really should probably not badger either Eldest or Eldest’s teacher with it, as he can go on to learn quite a bit of chemistry before he ever needs to deeply think about the issue.

  10. Hi XY,
    I agree that the way this is taught is fraught with approximations/cheats. Even at the J level, we can’t think of a single electron in a single orbital – the 3P2 ground state is still 5-fold degenerate – then you can split them into the mj states with a magnetic field.

    Unfortunately I don’t know a good but rigorous way of teaching this. The one-electron picture is fairly straightforward, and lets people just count electrons without worrying too much about how they interact, so everyone learns it that way. It’s when you get to spectroscopy that it all breaks down, since you need to talk about states and not orbitals.

    Looking forward to the next chemistry conundrum!

  11. What Josh is getting at is that the value of ml, ms is only defined with respect to the magnetic field direction. If there is no magnetic field, then all three states are equivalent. At that point, what is important is the interaction of the electrons with each other. The orientation of the spins is easy to figure out, they add algebraically to maximize the total electron spin (e.g. go in parallel to each other until the p or d subshell is half filled and then start to pair up.

    As far as the orbital angular momentum goes what is important is not the sign of for each electron ml, but the orientation of the orbital with respect to the orbitals of the other electrons in partially filled subshells. Hunds rule holds that the absolute value of the sum of the mls be a maximum. The vector model is a good crutch and indeed where Hund’s rules were first formulated


    What this means is that the question is meaningless because it assumes an absolute orientation in space with a privileged z axis.

  12. You are both wrong (and right) at the same time, the order does not matter and is arbitrary in the absence of magnetic field.

  13. The order maters because both the spin and the orbital angular momentum of the electron generate magnetic fields and the interaction of the electrons magnetic fields affect the energy level. In particular Hund;s rules are consequences of the interactions between the fields generated by orbital and spin angular momentum

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