# Is there a physical explanation behind the derivation of spectroscopic term symbols?

In deriving the spectroscopic term symbols, one method seems common among several sources, the clearest presentation for me being located here, a derivation of the term symbols for a $d^2$ ion.

So, the execution of this method seems pretty straightforward. My confusion is in why it works. It currently seems rather arbitrary (a $L=4$ G term symbol subtracts a microstate from eg. $L=0$). Stumbling blocks for me are:

### 1. Some singlet microstates seem redundant

http://sli.mg/j4t7mg.png

If I look at, for example, the wiki diagram for LS coupling by vector addition, I would assume S = 0 and L is equal to 1 in both of the above microstates.

### 2. Some higher multiplicity states also seem redundant

From the same vector addition diagram above, these top row microstates should be equivalent to the bottom row microstate cross-diagonal from it. As far as I have understood, the spectral term takes the absolute value of L and does not distinguish between -L and +L (eg. L=1 and L=-1 both yield P term symbols), so $|-L-S|=|L+S|$.

Alternatively, I have also considered that maybe the absolute value is only taken on L and not S. In this case, $|-L|+S=|L|+S$. Then the microstates would not be equivalent cross-diagonally but instead horizontally (top row microstates equivalent to each other and bottom row microstates equivalent to each other).

### 3. Not all term symbols for the potential microstates are a solution.

$\Rightarrow$ $^3D_3$, $^3D_2$, $^3D_1$ microstates vanish in the derivation

After the method was so thorough in including all the potential microstates, some of these microstates turn out to be nonexistent or physically not relevant.

I am interested to know roughly why this method for the derivation of term symbols is physically logical (why it works), more specifically why the above states are not redundant, and most especially why certain spectroscopic terms / electron microstates are not relevant. I'm afraid the question may be too open ended, so if I can clarify any further, or there's a concise reference, please let me know.

Thank you for your time!

• There isn't a one-to-one correspondence between those orbital diagrams and terms which makes it very hard to discuss the matter in this fashion. For example for the very first two diagrams you posted, the actual terms arise from linear combinations of those two orbital microstates (perhaps plus more, I'm not sure). – orthocresol Dec 31 '16 at 17:20
• The terms that don't appear are not "physically unimportant" but rather "physically forbidden" - they violate the Pauli exclusion principle. Again those orbital diagrams aren't very useful to find out why. You need to construct the wavefunctions as linear combinations of those orbital microstates. See for example (a simpler $\mathrm{p^2}$ case - $\mathrm{d^2}$ is more complicated but same idea) Pauli-forbidden term symbols for atomic carbon (btw, I don't think this is too open-ended, so don't worry about that.) – orthocresol Dec 31 '16 at 17:24
• Thank you for this link. I don't fully understand the spatial construction of microstates at a glance, but intend to clarify this later when I have had more time to look at it, as I am currently been torn away from the computer for tonight. Is a term symbol only allowed if every conceivable microstate corresponding to it is not Pauli forbidden (why $^3D$ is not allowed because one microstate for that term would involve a doubly occupied $m_l=1$, from your post) – Blaise Dec 31 '16 at 17:37
• also have a look at how this calculation is done chemistry.stackexchange.com/questions/63174/… and some explanation also at chemistry.stackexchange.com/questions/54721/… which are two examples of how to do these calculations. – porphyrin Dec 31 '16 at 18:10