Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2021 / Paper 2A / Section A / 15

The terms are ${}^3{\Sigma }_g^{-}$, ${}^1{\Delta }_g$ and ${}^1{\Sigma }_g^{+}$. Closed molecular orbital shells contribute no net spin angular momentum or axial orbital projection, so only the two equivalent ${\pi }_g$ electrons need be coupled. Their individual projections are $\lambda =\pm 1$.

Putting both electrons in the same projection gives $|\Lambda |=2$ and forces opposite spins by the Pauli exclusion principle, yielding ${}^1{\Delta }_g$. With one electron in each projection, the spin-adapted symmetric and antisymmetric spatial combinations give a singlet $\Sigma$ and a triplet $\Sigma$. Their reflection parities are $+$ and $-$, respectively. Both open-shell electrons are gerade, so their product is gerade. The dimensions $3+2+1=6$ exhaust the $\left(\genfrac{}{}{0ex}{}{4}{2}\right)=6$ microstates. By Hund's rules, the triplet is the lowest term, and ${}^1{\Delta }_g$ is the lower singlet.

The terms are ${}^1{\Sigma }_u^{+}$, ${}^1{\Sigma }_u^{-}$, ${}^1{\Delta }_u$, ${}^3{\Sigma }_u^{+}$, ${}^3{\Sigma }_u^{-}$ and ${}^3{\Delta }_u$. The active configuration is ${\pi }_u^3{\pi }_g^3$. Each nearly filled $\pi$ shell has the angular-momentum possibilities of one hole, so coupling two distinguishable hole sets gives $\Lambda =0$ or $|\Lambda |=2$ with $S=0$ or $1$.

For $\Lambda =0$, the two arrangements of opposite projections combine with either reflection parity, hence both ${\Sigma }^{+}$ and ${\Sigma }^{-}$. Unlike the equivalent-electron case in part (a), neither spatial combination is excluded for either spin: the holes belong to different orbital shells. The inversion parity is $u^3{g}^3=u$. Counting degeneracies gives $1+1+2+3+3+6=16$, equal to the $4\times 4$ choices of the two holes.

Only ${}^1{\Delta }_g\to {}^1{\Delta }_u$ is electric-dipole allowed within the listed terms. For a diatomic molecule, the electric-dipole selection rule requires $\Delta S=0$, $g\leftrightarrow u$, and $\Delta \Lambda =0,\pm 1$. The lower singlet of part (a) is ${}^1{\Delta }_g$. Every candidate from the same configuration fails the inversion rule, while the ${}^1{\Sigma }_u^{\pm }$ terms of part (b) require $|\Delta \Lambda |=2$. Triplet destinations fail the spin rule. The surviving $\Delta \to \Delta$ transition uses the axial dipole component, so it is a parallel electronic transition. Weak spin-orbit or other intensity-borrowing transitions are outside this leading electric-dipole classification.