Term symbol of a molecule in an excited state

[Aleolomorfo]

Homework Statement

Considering the molecule of $C_2^+$ in an excited state with valence configuration $2\sigma_g^2$ $2\sigma_u^2$ $1\pi_u^2$ $3\sigma_g^1$. Finding all the possible term symbol.

Homework Equations

Term symbol $^{2S+1}\Lambda$
$L=0 \rightarrow \Sigma$
$L=1 \rightarrow \Pi$
$L=2 \rightarrow \Phi$

The Attempt at a Solution

The problem is solved but I have some doubts about the solution. First of all, the term symbol are: $^2\Delta_g$, $^2\Sigma_g$, $^2\Sigma_g$, $^4\Sigma_g$. This is ok.

First, If I want to find all the possible configuration with the above valence configuration I have to calculate:
$$\binom{\text{maximum number of electrons which can be "host" in the shell}}{\text{electrons in the shell}}$$
Not considering the full shell ($\pi$ state can host 4 electrons, instead $\sigma$ states only 2):
$$\binom{4}{2}+\binom{2}{1}=8$$
But in the solution there is written that the possible states are 12, and I do not understand why.

Second, I do not understand how to calculate the degeneracies of each term. From my perspective should be $(2S+1)(2\Lambda+1)$, so $^2\Sigma_g$ should be $2\times (2\times 2+1)= 10$, but in the solution there are only 4 states.

I need some help to sort out my ideas, thank you in advance.

 

[DrClaude]

Not considering the full shell ($\pi$ state can host 4 electrons, instead $\sigma$ states only 2):
$$\binom{4}{2}+\binom{2}{1}=8$$
But in the solution there is written that the possible states are 12, and I do not understand why.

Is adding the two binomial coefficients the right thing to do?

Second, I do not understand how to calculate the degeneracies of each term. From my perspective should be $(2S+1)(2\Lambda+1)$, so $^2\Sigma_g$ should be $2\times (2\times 2+1)= 10$, but in the solution there are only 4 states.

I don't understand what you are doing here. And what is $\Lambda$ for a $\Sigma$ state?

 

[Aleolomorfo]

Is adding the two binomial coefficients the right thing to do?

No, I have to multiply, thank you.

I don't understand what you are doing here. And what is ΛΛ\Lambda for a ΣΣ\Sigma state?

I see it is a bit confusing, but now I have understood reading again my book, thank you anyway