Lowest energy levels of the $Cr^{3+}$ ion

Ok i've been stuck on this since yesterday. I've checked theory and still can't get through it. Also, I can't find relevant examples online or on my book (Bransden, Physics of Atoms and Molecules). Please help me before i set my house on fire.

The exercise text:

The electronic configuration with lowest energy of the $Cr^{3+}$ ion is [Ar]$3d^3$. Using the LS coupling scheme and Hund's rules, find the quantum numbers $L$ and $S$ of the four spectral terms with lowest energy among those which are generated by the above electronic configuration.

What i understood:

• Summing over spin we get two possible values of $S$: $\frac{3}{2}$ and $\frac{1}{2}$. First Hund Rule implies the ground state will be labelled by $S_{gs}=\frac{3}{2}$.

• Summing over $L$ we get six possible values: $6,5,4,3,2,1$ and $0$. Since the first Hund Rule requests $S_{gs}=\frac{3}{2}$ and Pauli Principle, $L_{gs}$ couldn't possibly $6,5$ or $4$. The first available $L$ value is $3$.

For these reason the ground state term will be $F^4$. I need three more.

How would i proceed here if things were manageable:

I would write down the microstates table and extract minor matrices like showed here. Thing is that we have 120 possible microstates. This exercise is just the first point of a really articulated 2 hours written examination. This couldn't possibly be the route.

Other ideas:

Always by use of the first Hund's rule I expect the other terms to come out from $D^4$,$P^4$, $S^4$ before stirring on term with $S=\frac{1}{2}$ [[This is not true]]*. But how can i know which terms are permitted by Pauli Principle without tabling the microstates?

An heartfelt thanks to whomever will help me with this.

Example: Carbon*

The electronic configuration of $C$ is [He]$2s^2 2p^2$. I discard the full $2s$ subshell, it won't contribute to $L$ and $S$. Adding up spin and orbital angular momentum i get a pool of possible spectral terms: $D^3,D^1,P^3,P^1,S^3,S^1$. Moreover i have only 15 possible microstates. Studying the microstates table i discover that Pauli would forbid the terms $D^3,P^1,S^1$. [[How do i proceed when this table get huge?]]

The remaining possible terms are $D^1,P^3,S^1$. I can tell by Hund Rules that $P^3$ will be the ground state's spectral term but can't say anything about $D^1$ or $S^1$ ordering besides they are up in energy scale above $P^3$.

*=edit 8/5

• And you have read through this hyperphysics.phy-astr.gsu.edu/hbase/Atomic/Hund.html#c1 – user163104 Aug 4 '17 at 19:14
• Just gave it a quick read but it looks like a Hund's rules recap! Still don't get how i can subtract the states Pauli's forbidden without working out explicitly the microstates table. Sure thing i'm missing something, it's maybe under my nose, but could you argument? Thanks :)! – deppep Aug 4 '17 at 19:28
• So it is a quartet. Anything else you need? – Pieter Aug 5 '17 at 12:03
• Hi pieter! $F^4$ is definitely a quartet but the exercise text seems to be asking for four spectral terms, so i need three more. But it could be i'm misunderstanding the text at this point. Let me know what do you think. – deppep Aug 5 '17 at 12:28
• Ok, I just interpreted this as "four lowest states". Hund's rules do not really say anything about ordering of spectral terms. In fact, $^2$G and $^4$P are quite close in energy. – Pieter Aug 6 '17 at 0:24