[blaisem]

Hello everyone.

Hund's first rule states that the maximum multiplicity of electrons is lowest in energy.

Specifically, in Wikipedia, two reasons for the first rule are supplied.

1. Electron-electron repulsion is minimized by spatial separation due to occupying different orbitals.

Makes sense.

2. Electrons align their spin to reduce nuclear-screening effects.

Here I am confused, and I don't have access to the textbook (Levine, Quantum Chemistry 1991) that Wikipedia cites as its source.  The Wiki page goes so far as to say that the traditional belief of alleviation of spatial repulsion is subordinated by relaxation of nuclear-screening effects.  It seems most places explain the rule through the traditional reasoning, but I haven't found a good explanation for how parallel spins reduce screening effects.

Does anyone have an idea how parallel electron spins reduce nuclear-screening effects?

Hund's second rule states that, for a given multiplicity, maximization of the orbital angular momentum L is favored.

On this forum, Dan provided a very nice analogy in 2006 of cars in a roundabout, where by traveling in the same direction, they experience less repulsion with each other.  This was in response to a question on Hund's first rule, but it seems more like an explanation of Hund's second rule.  Even so, regarding Hund's second rule, I don't understand why a large, "positive" angular momentum is preferred to a large, "negative" angular momentum.  If a d² subshell were filled into the -2, -1 orbitals, the electrons are still "traveling together in the same direction", just as if they had been placed into the +2, +1 orbitals.  There must be some other effect explaining why a large, positive angular momentum is preferred.

Does anyone here know why we fill in orbitals corresponding to positive magnetic quantum numbers before orbitals with negative magnetic quantum numbers?

Hund's 3rd rule: |L-S| governs less than half-filled subshells.  |L+S| governs subshells more than half filled.

I have read from Hyperphysics the reasoning behind this rule is due to spin-orbit coupling.  The energy of spin-orbit coupling,

ΔH = c(L·S), where c is an always positive coefficient. (sorry my LaTeX doesn't seem to be working :/)

and

L·S = 0.5 (J² - L² - S²)

In order to minimize ΔH, L·S must be minimized, which means J must be minimized.  This leads me to believe that L-S is used for less than half-filled subshells because L and S have the same sign in this case, wherefore J is minimized through subtraction.  When the subshell is more than half full, L and S must have opposite signs, so J is minimized via L + S.

The problem with this reasoning, of course, is that L and S always have a positive sign as dictated by the first two rules.  So, I don't see how either L or S becomes negative after a shell has been half-filled, mandating |L+S|.

Does anyone know the reasoning behind whether J is minimized via |L-S| or |L+S|from the third rule?

I appreciate you reading my question.  Thanks for any help you can provide!

[Corribus]

Regarding Hund's First Rule:

I don't particularly like Wikipedia's explanation either.  A far clearer explanation to my mind is provided here:

http://chemwiki.ucdavis.edu/Inorganic_Chemistry/Electronic_Configurations/Hund's_Rules

If you have additional questions on this article, please post a follow-up.

Regarding Hund's Second Rule:

What matters is the total (scalar) angular momentum, not the direction.  The negative and positive signs refer only to the direction of the angular momentum, not the magnitude.  The direction is furthermore arbitrary (except in, say, a magnetic or electric field).  So is the spin direction, incidentally.  By convention we usually draw the first electron in each orbital as "up" (positive spin).  However we could just as easily draw it "down".  It makes no difference - in the absence of an external EM field, the energy is the same, if only because molecules/atoms are rotating with respect to the lab frame anyway.  "Up" and "down", in other words, is artificial.  What matters is the relative momentum vectors of the various electrons in the system, and hence their sum total.

Regarding Hund's Third Rule:

Introductory physical chemistry texts don't often go into detail about the origin of this rule.  It's one of the few cases in p-chem where students are encouraged just to memorize something without looking too hard at it.  Being honest, I haven't really thought about it that much, either.  I seem to recall from a long time ago someone offering a reasonably simple explanation related to the fact that when the shell is more than half full, it's easier to visualize the system as an interaction between the spin and orbital momenta of holes rather than electrons, in which case the energetic stabilization term is reversed in sign.  This would be, I presume, because the spin angular momentum of a single hole would be opposite in sign compared to the spin angular momentum of a single electron.  Taking as an example - the three p-orbitals.  A situation with 1 electron and 5 electrons are functionally similar, except that one has a single electron and one has a single hole.  All things being equal, the total spin angular momentum of the 1 electron system would be opposite in sign to whatever the total spin angular momentum of the 5 electron system is.  So the expectations for Hund's rules would be switched.  You can kind of see this if you draw out all the microstates of the 1-electron and 5-electron configurations: the everything is pretty much changed in sign in the latter case.  I admit I haven't carried that logic through to a derivation of the rule, but I suspect it is related somehow to this premise. 

As a final word, Hund's rules make a lot more sense in the context of generated term symbols.  If you spend some time deriving term symbols for various electron configurations, you will quickly start to understand that the various L and S values represent vector additions of possible microstates.  I suspect some of your issues may stem from a confusion related to the distinction between total angular momentum and the angular momentum of the individual electrons.  I'm not sure of the context of your interest in this topic, but that was just a thought I had.

[blaisem]

Regarding Hund's First Rule:

So taking the explanation of the website you provided that electrons are like tiny magnets, I investigated magnetism and eventually reached the Exchange Interaction, which stated that, by having parallel spins, two electrons could not approach each other as closely as if they had antiparallel spins.  This reduces coulombic repulsion.

Is this the underlying explanation for Hund's first rule? It seems a little indirect that electrons "know" to align their spins as a response to the "threat" of coulombic repulsion.  I mean, it seems a little illogical for them to come to that conclusion based on an eventuality.  For example, if two antiparallel electrons wandered too close together, the relevant response of the system is to push the electrons back apart, not to tell one of them to flip its spin.  So, it just seems weird to me that a repulsive force leads to an effect that's not related to repulsion, namely spin flipping.  I am not questioning nature, I am just trying to communicate my understanding of the topic; I'm throwing it out there for verification.

But, if what I just said is correct, despite it being weird, then I think I understand this first rule.  Thank you for your help.

Regarding Hund's Second Rule:

I thought I remember hearing somewhere that we should always "fill" electrons from the highest possible orientation to the lowest possible orientation, eg. +2 -> +1 -> 0 -> -1 -> -2 h-bar units for the case of l = 2.

You are saying that filling in -2, -1, 0, +1, +2 would also be ok, as long as the magnitude would be the same?

And a question of a technicality that just occurred to me:

For p-orbitals, we have +1, 0, -1 orientations of the angular momentum.  Which of these orientations correspond to the Px, Py and Pz orbitals, or is the designation arbitrary?  For example, in a p² system, how does filling in +1, then -1 (skipping 0), raise the energy of the system when the corresponding orbitals are orthogonal, and the electrons should never have the ability to come into contact anyways?

Sorry for introducing in another question like that abruptly.

Regarding Hund's Third Rule:

I have encountered term symbols in two courses so far, but only had to derive them in one.  I suppose I haven't had all that much experience with them.  But I like your definition of addition of microstates; I hadn't thought of it that way before.

I looked around some more and I found this website that basically says that a lower J is lower in energy because electrons are negatively charged, while higher J is lower in energy for "electron holes" which are positively charged.  I am not sure now how J and charge interact to determine the system's energy.  I might look more into this tomorrow.

Thanks for your detailed reply so far, and if you have anymore insight, I would appreciate your time.

Incidentally, the website I linked above made the statement that a larger L leads to more lobes in the electron wavefunction, which leads to lower electron repulsion.  He follows that with a remark on the positve / negative J.  After that, he burns us chemists pretty bad.  I had to laugh at the randomness of the remark.  Anyways, if I run into more info on the lobe theory, I'll write it in here.

[Corribus]

blaisem, I will be happy to answer your questions.  However, I will be away for much of tomorrow.  If nobody else answers them in the meantime, I will try to get you my thoughts when I get back.

[Corribus]

I found a little unexpected time.  Here is a response to your last post:

It seems a little indirect that electrons "know" to align their spins as a response to the "threat" of coulombic repulsion.  I mean, it seems a little illogical for them to come to that conclusion based on an eventuality.  For example, if two antiparallel electrons wandered too close together, the relevant response of the system is to push the electrons back apart, not to tell one of them to flip its spin.  So, it just seems weird to me that a repulsive force leads to an effect that's not related to repulsion, namely spin flipping.  I am not questioning nature, I am just trying to communicate my understanding of the topic; I'm throwing it out there for verification.

You are thinking too classically.  Electrons can't just occupy any space they want - their orbits, spins, momenta, energy and so forth are quantized and can only take on particular values.  Certain combinations of values are also forbidden.  Moreover, electrons don't have single, defineable positions.  They wont "wander close to each other".  We can only speak in terms of probability densities, average positions, and the like.

In quantum chemistry there are things you just have to accept, even when they don't always make intuitive sense.  It is common to search for classical analogs of quantum phenomena, and they can helpful tools (such as the nature of electron "spin"), but we must not look at them too closely because they WILL fall apart under scrutiny.

I am sorry this isn't a more helpful answer.

I thought I remember hearing somewhere that we should always "fill" electrons from the highest possible orientation to the lowest possible orientation, eg. +2 -> +1 -> 0 -> -1 -> -2 h-bar units for the case of l = 2.

You are saying that filling in -2, -1, 0, +1, +2 would also be ok, as long as the magnitude would be the same?

Yes.  This is why we refer to them as microstates.  In the absense of an external field, the five l = +2....-2 orbitals are degenerate.  They have exactly the same energy and there is no way to distinguish which of these states an electron is in at a given time.  In fact, because they all have the same energy, it might be said the electron is in all of them at once.  Remember, this is just the projection of the momentum along one axis, not the total orbital momentum.  The total orbital momentum is 2, regardless of what d-orbital the electron is in.  The only thing that changes is the proportion of that total momentum oriented in the z-direction.

But even ignoring that mind-twisting idea, if you compare just the +2 and -2 mL values, this just refers to the projection of the orbital angular momentum along an arbitrarily defined (and freely rotating) z-axis.  We can define the z-axis however we want, so while we might consider +2 as facing up and -2 as facing down, we could just as easily say -2 is facing up and +2 is facing down.  The distinction is artificial.  Even more, we don't even have to pick the z-axis, and only do so because the mathematics are simplified.  We could easily deal with the project along the y-axis or x-axis (or any other axis), which which case the projection along THAT axis could be quantized and the other two remain uncertain.  The important thing is that orbital angular momentum is quantized, and so too is the momentum along a single axis, with the momentum along the other two, perpendicular axes having complete uncertainty.

For p-orbitals, we have +1, 0, -1 orientations of the angular momentum.  Which of these orientations correspond to the Px, Py and Pz orbitals, or is the designation arbitrary?  For example, in a p2 system, how does filling in +1, then -1 (skipping 0), raise the energy of the system when the corresponding orbitals are orthogonal, and the electrons should never have the ability to come into contact anyways?

None of those orientations correspond to Px, Py and PZ.  Well, there isn't a simple correlation, anyway.  This is because the m_L = +1 and -1 states are complex and thus don't have meaning in real space.  In order to generate real, directed orbitals from the wavefunctions, we have to take linear combinations of the +1 and -1 orbitals so that the imaginary terms cancel out.  If I recall, the Px orbital is the "plus" combination and the Py orbital is the "minus" combination, and there are some normalization factors as well.  Because the m_L = 0 state has no imaginary component, it exists in real space already, and this is the Pz orbital.

[blaisem]

I found a little unexpected time.  Here is a response to your last post:

Thank you as always, Corribus, for your thorough and lucid answers. Have a good rest of your weekend!