I'm reading Gerhard Herzberg's "Atomic spectra and atomic structure" (can be accessed on archive.org, the relevant part is from p.82-87) and I don't quite grasp how to determine the possible values of the total orbital angular momentum $L$.
To determine the possible values of $L$ for a two valence electron atom I just add both $l_i$ and decrease this by 1 until I reach the absolute value of the difference of $l_i$. For example: for $l_1=1$, $l_2=1$ the total orbital angular momentum quantum number $L$ can be $l_1+l_2=2$, $1$, $|l_1-l_2|=0$ so the term symbols are D, P, S.
Herzberg gives an example for three p-electrons as well, as I understand it you at first only consider two electrons giving $L_{pp}=2,1,0$ as above, and then add and substract the other electron's $l$ to all possible $L$ values resulting in the following possible values for $L_{ppp}=3,2,1,0$ giving S, P, D, F term symbols (it's shown on p.87). I'm not sure I got this right, Herzberg writes:
"the vector addition can be carried out simply by combining the $l$ values of two electrons and then combining each of the resulting L values with the $l$ of the third electron".
What does he mean with combine each of the L values with the $l$ of the third electron? He gives the following possible values of $L_{ppp}=0,1,1,1,2,2,3$.
If I add 1 to each of the $L_{pp}$ I get $L_{pp+1}=1,2,3$ now when I substract 1 from $L_{pp}$ I get $L_{pp+1}=-1,0,1$ ignoring the negative ($L$ can never be negative) and combining $L_{pp}, L_{pp+1}$ and $L_{pp-1}$ I get $L_{ppp}=0,0,1,1,1,2,2,3$
Am I right with the assumption to just add and substract the $l$ of the third electron?
