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### What can visual observations of pi-bonding MOs explain about LCAO expansion coefficients?

I've thought that in N2, its symmetric so the expansion coefficients would be equal from the pi-bonding MO Yes, that's right; symmetric molecules have symmetric MOs which have symmetric coefficients ...
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I think maybe there is some confusion in terminology. Maybe these notes will help you: When we take spin into account, we combine the spatial orbital of the ith electron $\psi_i(\mathbf{r})$ that ...