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 / b / Solution

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.