Determining Ground State in L-S Coupling Scheme for (3d)4

[secret2]

When doing L-S coupling scheme for (3d)4, I am asked to find the ground state. I know that the first step, according to Hund's rules, is to find the the S with the largest value (S, L are total values, s, l are individual values). It is obviously S=2. Next, I should proceed to find the largest possible value of L. And here is my question: why can't we choose L=4*2=8? Why should we choose L=2?(Please continue reading)

The book says that anything larger than L=2 would have values of (m (sub L)) larger than 2, which is prohibited by the Pauli exclusive principle because no two electrons can have same pair of (m (sub l)) and (m (sub s)). However, why do we have to throw away the whole set of states with a specific L just because some (m (sub L)) are illegal?

 

[dextercioby]

Well,each time u have an "L",u have the "2L+1" values of "m" to go along...Now if L=3,then in principle m_{l}=3 (okay?).But if u cannot have m_{l}=3,then it would make no sense in including the L=3 (which automatically would induce m_{l}=3)...

Daniel.

 

[R0man]

In the L-S coupling scheme, the total angular momentum (J) is given by the vector sum of the orbital angular momentum (L) and the spin angular momentum (S). In the case of (3d)4, the maximum possible value of S is 2, as you have correctly identified. However, the maximum possible value of L is not simply 4*2=8, but rather it is limited by the number of electrons in the system.

In the (3d)4 system, there are 4 electrons, each with a spin of either +1/2 or -1/2. This means that there are only 2 possible combinations of spin states for the 4 electrons: ++--, --++, or any combination in between. In order to satisfy the Pauli exclusion principle, which states that no two electrons can have the same set of quantum numbers, the values of (m (sub s)) must be different for each electron. This means that for a given value of S, there can only be a certain number of allowed values of (m (sub l)).

In the case of L=8, there would be 5 allowed values of (m (sub l)): -8, -6, -4, -2, 0. However, since there are only 4 electrons, one of these values would have to be repeated, violating the Pauli exclusion principle. Therefore, the maximum possible value of L in this system is actually 4, which corresponds to the allowed values of (m (sub l)) for a total of 4 electrons: -4, -2, 0, 2.

In summary, we cannot choose L=8 in the L-S coupling scheme for (3d)4 because it would lead to a violation of the Pauli exclusion principle. We must choose the maximum allowed value of L, which in this case is 4, in order to satisfy the rules of quantum mechanics.