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I want to ask a question about the symmetry for $\pi$ orbitals.

We were learning today about the deduction of an MO diagram for a typical $\ce{M-M}$ system with $\ce{D_{\infty h}}$ and I was confused on the labelling of the $\pi$ orbitals.

When labelling the $\ce{dz^2}$ orbitals, the following checklist was produced:

orbitals along the bond $\rightarrow$ $\sigma$

symmetry change through $i$ $\rightarrow$ g

no phase change for $\sigma_{v}$ $\rightarrow$ +

overall symmetry $\sigma_{g}^+$

but when we were labelling the $\ce{d_{yz}}$ and $\ce{d_{xz}}$ orbitals, the following information was given, without further explanation:

no $+$ or $-$ label for $\pi$ orbitals

I worked out that the $+$ or $-$ relates to how the phase changes after reflection in the $\sigma _{v}$ but still couldn't work out why I can't apply the signs to the $\pi$ MOs.

Why is there no $+$ or $-$ notation for these $\pi$ MO's?

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vik1245
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  • Pick the yz plane as $\sigma_v$. What happens to the phase of $d_{yz}$ when $\sigma_v$ is applied? What happens to $d_{xz}$? – Tyberius Oct 22 '19 at 21:38
  • @Tyberius for both I think there is no change in phase. If I am correct, can't I just write $+$ anyway? – vik1245 Oct 22 '19 at 22:21
  • You are right for the $d_{yz}$, since the mirror plane is in the same plane as the orbital. But for $d_{xz}$ pair, the mirror plane bisects each orbital, flipping the lobes around the z-axis. So each of the π orbitals have different phase behavior, so in total it can't be + or -. The same is true for any of the nonsigma orbitals, as can be seen in the $D_{\infty h}$ character table. – Tyberius Oct 22 '19 at 22:37

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