Hund's rules and Pauli's principle

In summary, there are two methods for applying Hund's rules to determine the ground state of an electron configuration. Method 1 involves determining all possible states and selecting the state with the highest total S and L, while Method 2 involves filling up the possible m_l states in a specific order to find the values for L and S. The two methods are equivalent because they both satisfy Pauli's principle and the triangle rule for determining L.
  • #1
Reignbeaux
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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 [itex] m_l [/itex] states beginning with the highest [itex] m_l [/itex]. First, spins with [itex] m_s = \frac{1}{2} [/itex] are assigned; when all [itex] m_l [/itex] are occupied once, [itex] m_s = -\frac{1}{2} [/itex] are assigned again beginning with the highest [itex] m_l [/itex]. Now one can calculate [itex] m_L = \sum{m_l} [/itex] and [itex] m_S = \sum{m_s} [/itex]. Now comes the confusing part:
The ground state simply has [itex] L=m_L [/itex], [itex] S=m_S [/itex].

I don't really get why the two methods are equal. Why is [itex] L=m_L [/itex]? 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 [itex] L=m_L=1+1=2 [/itex] is antisymmetric and [itex] S=m_S=\frac{1}{2} + \frac{1}{2}=1 [/itex] is symmetric. So maybe this is due to a general property and the method works because it satisfies Pauli's principle?
 
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  • #2
Reignbeaux said:
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?
 
  • #3
DrClaude said:
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
 
  • #4
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##.
 
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  • #5
DrClaude said:
[..] 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.
 
  • #6
Reignbeaux said:
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##.
 
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  • #7
Thank you so much, I can see the connection now. Perfect!
 

Related to Hund's rules and Pauli's principle

1. What are Hund's rules?

Hund's rules are a set of three principles that describe the arrangement of electrons in an atom's orbitals. These rules state that electrons will fill orbitals of the same energy level with parallel spins before pairing up, and that they will occupy different orbitals within the same subshell as much as possible.

2. What is Pauli's principle?

Pauli's principle states that no two electrons in an atom can have the same set of quantum numbers. This means that each electron must have a unique combination of energy level, orbital, and spin quantum numbers.

3. How do Hund's rules and Pauli's principle relate to electron configurations?

Hund's rules and Pauli's principle are both important in determining the electron configuration of an atom. Pauli's principle determines the overall arrangement of electrons in different orbitals, while Hund's rules dictate the specific order in which electrons fill those orbitals.

4. Why are Hund's rules and Pauli's principle important in chemistry?

These principles are important in chemistry because they help explain the behavior of atoms and their electrons. They also play a crucial role in understanding the properties and reactivity of elements and compounds.

5. Are there any exceptions to Hund's rules and Pauli's principle?

Yes, there are some exceptions to these rules. For example, in transition metals, the 4s orbital may fill before the 3d orbital due to the lower energy of the 4s orbital. Additionally, some atoms may have half-filled or fully filled subshells due to the increased stability of these configurations.

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