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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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What does the χ (chi) symbol represent in a Slater determinant?

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 ...

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