Hund's Rule Question

[Old Guy]

I'm certainly missing something here. The problem is to find the ground state L and S for Gd, with a configuration given as [Xe]6s^2,4f^7,5d^1.

I start by calculating Smax. I'm figuring you get 1/2 for the 5d^1 electron, plus (since the f level has room for 14 electrons, all 7 can have +1/2 spin) 7/2 give Smax = 4.

How do I get L? It would seem that Lmax would be 4 for the 5d^1 electron and 3 for the others, making a total of 7. The book says it should be 2, and I can't see how, for example, the Pauli exclusion principle should get me there. Any help would be appreciated - I really want to understand the concept as much as deal with this problem. Thanks.

[OlderDan]

I'm certainly missing something here. The problem is to find the ground state L and S for Gd, with a configuration given as [Xe]6s^2,4f^7,5d^1.

I start by calculating Smax. I'm figuring you get 1/2 for the 5d^1 electron, plus (since the f level has room for 14 electrons, all 7 can have +1/2 spin) 7/2 give Smax = 4.

How do I get L? It would seem that Lmax would be 4 for the 5d^1 electron and 3 for the others, making a total of 7. The book says it should be 2, and I can't see how, for example, the Pauli exclusion principle should get me there. Any help would be appreciated - I really want to understand the concept as much as deal with this problem. Thanks.
The magnetic quantum numbers m_l have 2l+1 possible values ranging from -l to +l in integer steps. The s^2 has only 0, the p^7 has one of every m filled, so the net contribution from these is zero. The one remaining electron in the d^1 has a maximum value of 2, so it looks like 2 is it.