How to determine possible term symbols in case there are more than 2 electrons?

Question

I'm doing some exercises on $LS$ coupling during my self-study of atomic physics. I understand how to construct a table of possible microstates. I also understand how to map term symbols to those cells of microstates: the total wave function must be antisymmetric under exchange of both electrons. Combining the fact that the parity of the spatial wave function is $(-1)^L$, spin singlets yield an antisymmetric spin wave function and spin triplets yield a symmetric spin wave function then allows me to determine the possible term symbols.

Now how does this work when there are more than 2 electrons? I know the wave function must still be antisymmetric under exchange of any two electrons. My notes contain the following tables:

The first table is for a $2p^23s^1$ configuration, the second table for a $np^3$ configuration. I still understand how to figure out all possible microstates, but I can't seem to figure out which term symbols are allowed. Take for example the $M_L=0$, $M_S=3/2$ cell in the second table. I would say the spatial wave function is symmetric since $L=0$ and the spin wave function is symmetric as well since all spins are the same. So the total wave function is symmetric, which obviously can't be right. What am I doing wrong here?

Any help would be greatly appreciated!

Answer 1

You calculate the total orbital angular momentum $L$ of the combined system. For two electrons, e.g. both in the $p$ shell, that could take values from $|1-1| = 0|$ to $(1 + 1) = 2$ in steps of $1$. So you may have $L=0, 1,$ or $2$.

The actual proof is (I think) quite mathematical with Slater determinants, but the general rule of thumb is that:

For odd numbers for the total angular momentum $(L= 1,3,5, ...)$ the spatial wavefunction is antisymmetric upon particle exchange.

So for $L=0$ and $L=2$ the spatial wavefunction is symmetric, and you therefore need an antisymmetric spin part (singlet, total spin $S=0$. For $L=1$, you need a symmetric spin part (triplet, $S=1$). The spin values here are only valid for a $2$ level system.