Term symbols and Pauli principle

Question

Given a Silicium atom with $1s^22s^22p^63s^23p^2$ I have to write down all possible term symbols.

The solution says a possibility - apart from $^1S$, $^1D$ and $^3P$ - would be $^1P$, which - due to the Pauli principle - does not exist.

And here is my question: Why does $^1P$ not exist? Is there an easy way to see why $^1P$ can not exist, just by looking at the term symbol? Or maybe more general: Is there a way to look at possible term symbols and immediately say whether they can exists or not?

(I don't know much about theoretical quantum mechanics, the question was part of an experimental physics exam.)

Answer 1

In the special case of Silicium, you have two valence electrons in a p orbital; knowing that the total orbital angular momentum is L=1 (this is the meaning of letter P) the orbital wave function of the two electrons changes sign under parity (parity---> $(-1)^L$) and, since the electrons are two, it means that the spatial part is antisymmetric under electron exchange. Since the total state must be antisymmetric and the state is the product between the orbital wave function and the spin function, you can't use a singlet state (which is antisymmetric) and L=1 because in this way you end up with a symmetric total state. Note that singlet has S=0 and it corresponds to the number in the upper left side of the term notation (which actually is 2S+1). More generally, if you have more than two equivalent electrons, you should use Clebsch-Gordan coefficients to combine different angular momenta and then figure out whether the total wave function is spatially symmetric or antisymmetric. When you combine two l=1 angular momenta you can have L=0,1,2. The former and the latter are symmetric under electron exchange (and can be coupled with the singlet), while L=1 is antisymmetric and can be combined only with the triplet state.