Using Hund's rules to find ground state L, S, J

In summary: L?When you have the maximum L and maximum S, the ## \vec{J}=\vec{L}+\vec{S} ## still holds. However, when you get down to the case where L and S are not maximum, the ## \vec{J}=\vec{L}-\vec{S} ## holds. So in the case of Titanium, the minimum ## J ## is 3.
  • #1
astrocytosis
51
2

Homework Statement



Using Hund's rules, find the ground state L, S and J of the following atoms: (a) fluorine, (b) magnesium, and (c) titanium.

Homework Equations



J = L + S

The Attempt at a Solution



I'm having trouble understanding what L, S and J mean on a basic level. I read the textbook chapter multiple times but I'm still not grasping their physical significance and how they are related to s, l and j.

I think I need to look at the outer shell, so for Mg for example the outer shell is an s orbital which I know means L = 0. And J = L + S. But I'm unsure about how to find S. I think it might be 1/2 + -1/2 = 0 since there are 2 electrons and one must be spin up and the other spin down? This makes J = 0. So the answer would be 0, 0, 0. Is that correct? I didn't really make use of Hund's rules here.

Using this reasoning I would get S = 1/2, L = 1, J =1+1/2 for F and S = 1, L = 2, J = 3 for Ti. Is this the correct approach? Is there a good way to make sense of S, L, and J so I don't get so confused about what they represent?
 
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  • #2
Titanium has ## 3d^2 4s^2 ##. (I googled it.) The ## S=1 ## looks correct, but what is the maximum ## L ## that you can make? Then, to apply the 3rd rule, with the shell less than or equal to be half-filled, what is the minimum ## J ## ?
 
  • #3
I'm having trouble understanding how L can have more than one value. I thought that L = 2 for d shells and L = 0 for S shells always. Maybe it can be 3 since n = 4 for 4s2?
 
  • #4
astrocytosis said:
I'm having trouble understanding how L can have more than one value. I thought that L = 2 for d shells and L = 0 for S shells always. Maybe it can be 3 since n = 4 for 4s2?
In the d=2 orbital, ## l=2 ## for each electron. You are adding angular momentum: ## \vec{L}=\vec{l_1}+\vec{l_2} ##. With 2 electrons each of l=2, you can generate an ## L=4 ## state. The possible ## L ## states (for ## l_1=2 ## and ## l_2=2 ##) are ## L=4,3,2,1, \, and \, 0 ##. The topic that covers it in more detail is found under the category Clebsch-Gordon coefficients, and I think Racah coefficients, but if you read anything that gives a good summary of the addition of angular momentum, that should be sufficient for the purpose at hand. (Similarly you are adding ## \vec{S}=\vec{s_1}+\vec{s_2} ##. Finally, you take the ## L ## and ## S ## and get the ## \vec{J}=\vec{L}+\vec{S} ##. In all 3 cases, you are adding angular momentum. The process is a little bit odd, but good explanations of it can be found if you google it and read the wikipedia summary, etc.)
 
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  • #5
Ok, that makes sense; somehow I missed that L is just the sum of each electron's l. So for Ti S = 1, L = 4, and J = 1 since L = l + l = 0 + 0 = 0 for minimum J?
 
  • #6
astrocytosis said:
Ok, that makes sense; somehow I missed that L is just the sum of each electron's l. So for Ti S = 1, L = 4, and J = 1 since L = l + l = 0 + 0 = 0 for minimum J?
You want the minimum ## J ## (since the shell is half-filled or less) with the maximum ## L ## and maximum ## S ##. Since ## \vec{J}=\vec{L}+\vec{S} ##, with the way it works, when ## L=4 ## (which is the correct maximum L) and ## S=1 ## (which is the correct maximum S), the available J's are 5,4, and 3. The minimum of these is 3. (With this state comes an ## m_J ##, but they are not asking for that.) ## \\ ## Additional note: Here (for Titanium) we are basically working with the 2 ## 3d ## electrons since the ## 4s^2 ## forms a closed shell. The ## \vec{L}=\vec{l_1}+\vec{l_2} ## is first computed, and then the ## \vec{S}=\vec{s_1}+\vec{s_2} ## and then finally the ## \vec{J}=\vec{L}+\vec{S} ##
 
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  • #7
How do you obtain the other available J's? Also wouldn't the maximum L be 3 because of the way the d subshells fills up; there are 5 subshells and each subshell has a different l (from -2 to 2) so you would get 2(1) + 1(1) = 3 if there are 2 electrons in two different subshells?
 
  • #8
astrocytosis said:
How do you obtain the other available J's? Also wouldn't the maximum L be 3 because of the way the d subshells fills up; there are 5 subshells and each subshell has a different l (from -2 to 2) so you would get 2(1) + 1(1) = 3 if there are 2 electrons in two different subshells?
I am no expert on this, but also on a learning curve. A google (hyperphysics) shows the S=1 state is symmetric so the only allowed L's for S=1 are are L=1 and L=3 since these states are antisymmetric. (The total electron state must be antisymmetric). With the maximum L=3 and maximum S=1, the available J's are 4,3, and 2 and the minimum J=2 would be the ground state by Hund's 3rd rule. There is also, I think, the possibility of S=0 and L=4. Exactly how the d shells fill in is beyond my expertise in this area. Even the case of the p orbitals with two p electrons would be something I would need to research further.
 

Related to Using Hund's rules to find ground state L, S, J

1. What are Hund's rules?

Hund's rules are a set of three guidelines used to determine the ground state of an atom or molecule. They were developed by Friedrich Hund in the early 1920s and are based on observations of the electron configurations of different elements.

2. How do Hund's rules help us find the ground state L, S, and J?

Hund's rules help us determine the ground state by providing a systematic way to assign electrons to different orbitals. This allows us to determine the values of L (orbital angular momentum), S (spin quantum number), and J (total angular momentum) for the ground state of an atom or molecule.

3. What is the first Hund's rule?

The first Hund's rule states that for a given energy level, the ground state has the maximum number of unpaired electrons with parallel spins. This means that electrons will occupy separate orbitals with the same spin before pairing up with opposite spins.

4. What is the second Hund's rule?

The second Hund's rule states that if two or more orbitals of equal energy are available, electrons will fill them in a way that maximizes the total spin quantum number, S. This means that electrons will first occupy separate orbitals with the same spin before pairing up with opposite spins.

5. What is the third Hund's rule?

The third Hund's rule states that if two or more orbitals of equal energy are available and there are already electrons in some of these orbitals, the electrons will fill the empty orbitals in a way that minimizes the total energy of the atom or molecule. This means that electrons will first occupy the available orbitals with the lowest energy before moving to higher energy orbitals.

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