Past exam of the chemistry course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / 2 / b

End-on overlap produces $\sigma$ and ${\sigma }^{\ast }$ molecular orbitals; side-on overlap produces $\pi$ and ${\pi }^{\ast }$ molecular orbitals. Matching signs of the facing atomic orbital lobes give constructive interference and a bonding molecular orbital. Opposite signs give destructive interference and an additional nodal plane between the atomic nuclei in the antibonding molecular orbital.

Take the internuclear direction as $x$, and define both atomic $p_x$ functions as positive on their own right-hand side. Up to wavefunction normalization, the combinations areThe minus sign in the bonding $p_x$ combination follows from this common-axis quantum phase convention: the facing lobes of the two unmodified $p_x$ functions have opposite signs. It does not make that combination antibonding. Every $\pi$ molecular orbital also has a nodal plane containing the internuclear axis; the additional midpoint node distinguishes ${\pi }^{\ast }$ from $\pi$.

The labels apply when the molecule has an inversion centre, as a homonuclear diatomic molecule does. The label $g$ means $\psi (-\mathbf{r})=\psi (\mathbf{r})$, and $u$ means $\psi (-\mathbf{r})=-\psi (\mathbf{r})$, with the origin at that centre. In a heteronuclear diatomic molecule, inversion exchanges different atomic nuclei and these are generally not valid symmetry labels.

For the p-derived combinations above,Inversion both exchanges the two centres and changes the sign of each individual p atomic orbital. Applying both operations to each sum or difference gives these labels. The star denotes antibonding molecular orbitals, independently of parity: a bonding molecular orbital need not always be $g$.