Hund's rules confuse me!

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I would like to ask two questions about Hund's rules and L-S coupling:

1. Some textbooks state that when doing L-S coupling and applying Hund's rules, "The maximum values of S and L are subject to the condition that no two electrons may have the same pair of values for m(sub s) and m(sub l). I know this is because of the Pauli exclusion principle, but how does this requirement (m(sub s) and m(sub l)) really limit S and L when we are adding the angular momenta?

2. When we are trying to figure out the ground state of Sm (4f)6, why is it wrong to have L = Sum(l) = 6*3?

Finally, I've realized that in discussing Hund's rules and L-S coupling some texts tend to make explanations using symmetry consideration and the others tend to prefer the exclusion principle. Are they two different sets of explanations, or are they equivalent?
 
The explanations are equivalent. With regards to [tex]Sm 4f^6[/tex], then n=4, l=3, [tex]L=\sqrt{l(l+1)}\hbar[/tex] and I'm not sure how to interpret the 6.

With regards to your first question, the ms and ml states are simply the number of possible states at that level. When doing spin orbit coupling, (and this is where I get a little flakey), [tex]L_{z}[/tex] and [tex]S_{z}[/tex] no longer commute with the hamiltonian, but [tex]L^2 , J^2, S^2[/tex] do, and you have to use [tex]J^2 = (L+S)^2 = L^2 + S^2 + 2S.L[/tex]. I hope thats right. Should be I have an exam on it in the next fortnight!

C.
 

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