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I have been reading about the ways to determine the ground of state of an atom. There are three Hund's rules in determining which electronic state is a ground state. And the second rule says you need to maximize the orbital angular momentum while considering the symmetry problem of the total wave function. I know that you need either spin or spatial wave functions to be symmetric. For spin, it is either singlet (anti-sym) or triplet (sym). However, when it comes to s, p, d, f, spatial wave functions corresponding to different orbital angular momentum, how do you know which one is symmetric and which is not?

For example, carbon 1s2 2s2 2p2

Maximize spin: $S=1$ (triplet, symmetric)

Maximize $L$: $L=2$ or $L=1$

I know that $L=1$ is the correct answer but I don't know why is $L=1$ (p) is antisymmetric while $L=2$ (d) is symmetric. Are there general rules for this to determine this symmetric properties of spatial wave functions.

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Simply put, angular momentum eigenfunctions with total angular momentum quantum number $L$ will have well-defined parity $(-1)^L$.

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