I'm currently studying atomic term symbols. I wanted to try it on a simple atomic carbon with the electron configuration $1s^22s^22p^2$.
I know, that only open-shell electrons are involved in the term symbol classification, so that leaves us working with solely 2 electrons in $2p$ sub-shell.
First of all, we have ${6}\choose{2}$$= \frac{6!}{2!(6-2)!}=15$ possible microstates for the $1s^22s^22p^2$ configuration.
The microstates are listed in the following table:
And, if I understand it correctly, we can categorize them into several "subsets" labeled with the corresponding term symbols.
The atomic term symbol is defined as ${}^{2S+1}L_J$.
In our case, the possible values of S, L and J are following: $$\max(M_L) = \max(L) = +2 \Rightarrow L = 0,1,2$$ $$\max(M_S) = \max(S) = +1 \Rightarrow S = 0, 1$$ $$J = L+S, L+S-1, \ldots, |L-S| = 3,2,1,0$$
Considering possible values of $S$, multiplicity can be $1$ or $3$, i.e. singlet or triplet. Values of $L$ enable symbols $S,P,D$.
So, our presumed set of states is ${}^3D, {}^1D, {}^3P, {}^1P, {}^3S, {}^1S$.
${}^3D$ "contains" $S=1$ and $L=2$. In that situation, both electrons would have to be spin-up and positioned together in the orbital with $m_l = +1$. That is impossible, as it contradicts Pauli exlusion principle. I.e. the state ${}^3D$ does not exist.
${}^1D$ is possible ($L=2,1,0, S=0$), so we can compute the number of corresponding microstates $N$ with the formula $N = (2L+1)(2S+1) = 5\cdot 1 = 5$.
Using the same formula, ${}^3P$ is going to contain 9 microstates.
With the "substraction method" described in this video, we can arrive to the conclusion, that both ${}^1P$ and ${}^3S$ won't exist.
And finally, ${}^1S$ will contain the last one state.
Question
Now I know the possible term symbols, but I'm not sure, which microstates belong to them specifically.
It's clear, that microstates 13 and 15 will belong to ${}^1D$, but how can I determine it for the other states? ${}^1D$ should contain 3 more microstates with $S=0$ and $m_L$ being equal to -1, 0 and +1, but there are multiple candidates for every configuration.
So how could I distinguish between, e.g. microstates 7 and 9 or 8, 11 and 14?
I don't understand this point, as even $J=2$ for ${}^1D$, so it can't be used to distinguish among the "similar" microstates and to assign them properly.