Hund's rules and Pauli's principle

[Reignbeaux]

TL;DR Summary

Question on Hund's rule to determine ground sate of electron configuration.

So as I can see from the literature there are two "methods" on how to apply Hund's rules to determine the ground state of an electron configuration.

Method 1: One determines all possible states due to Pauli's principle (wave function must be totally antisymmetric) using angular momentum addition rules. Then one can select the sate with highest total S and L and can then also select J according to the 3rd rule.

So far, so good. This is quite lengthy, especially when one has to add multiple spins together, but it works and is plausible.
But one can also find this approach, for example on wikipedia:

Method 2: One fills up the possible $m_l$ states beginning with the highest $m_l$. First, spins with $m_s = \frac{1}{2}$ are assigned; when all $m_l$ are occupied once, $m_s = -\frac{1}{2}$ are assigned again beginning with the highest $m_l$. Now one can calculate $m_L = \sum{m_l}$ and $m_S = \sum{m_s}$. Now comes the confusing part:
The ground state simply has $L=m_L$, $S=m_S$.

I don't really get why the two methods are equal. Why is $L=m_L$? According to angular momentum addition rules, L could also take higher values than that.

I may have found a hint on the solution to this, but I'm not sure if I'm on the right track and also I don't know how to generalize this and kind of proof the equivalence of the two methods: When for example one looks at the Clebsch Gordan coefficients for 1/2x1/2 and 1x1 one can see that $L=m_L=1+1=2$ is antisymmetric and $S=m_S=\frac{1}{2} + \frac{1}{2}=1$ is symmetric. So maybe this is due to a general property and the method works because it satisfies Pauli's principle?

 

[DrClaude]

But one can also find this approach, for example on wikipedia:

I've never seen this before. Could you please give a link to the Wikipedia page?

 

[Reignbeaux]

I've never seen this before. Could you please give a link to the Wikipedia page?

You can find it here in section "Ground state term Symbol": https://en.m.wikipedia.org/wiki/Term_symbol

 

[DrClaude]

Ok, I see now.

Basically, instead of listing all microstates to find all the term symbols, and then extracting out the one with highest S and highest L, the method gets you to construct the microstate with highest S (since you are putting the electrons one at a time maximizing spin while respecting the Pauli exclusion principle), and the build up of the $m_l$ from high to low allows to figure out what is the highest $L$ compatible with that highest $S$.

 

[Reignbeaux]

[..] and the build up of the $m_l$ from high to low allows to figure out what is the highest $L$ compatible with that highest $S$.

Thank you for your reply. But how can one see that $L = m_L = \sum{m_l}$ of all is the highest L compatible (yielding totally asymmetric wave function) with S? This doesn't seem trivial, at least not to me.

 

[DrClaude]

Thank you for your reply. But how can one see that $L = m_L = \sum{m_l}$ of all is the highest L compatible (yielding totally asymmetric wave function) with S? This doesn't seem trivial, at least not to me.

That actual equation to consider is $M_L = \sum m_l$ (triangle rule). The corresponding $L$ is then maximum value of $M_L$, since $L = -M_L, -M_L +1, \ldots, M_L$.

This is not different from the procedure you use when you list out all the microstates. For a given maximum $M_S$, you have a corresponding value of $S$, then, for that value of $S$, you find the possible value of $L$ by taking the maximum $M_L$.

 

[Reignbeaux]

Thank you so much, I can see the connection now. Perfect!