Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / 49 / a

For the ordinary spatial symmetries here, a symmetry operator is a unitary operator $U$ preserving the Hamiltonian:It maps an energy eigenstate to another state with the same energy and preserves the quantum dynamics. More general quantum symmetries can be antiunitary; time reversal is an important separate case.

An ordinary molecular point group consists of the identity, proper rotations $C_n$, and, when present, reflections $\sigma$, inversion $i$, and improper rotations $S_n$. Each operation leaves the nuclear geometry and chemical labels equivalent, so only identical nuclei can be interchanged. Translations are not elements of the point group of a finite isolated molecule. The molecular symmetry operations determine how electronic and vibrational coordinates transform.

The spin isomers of hydrogen illustrate how nuclear spin statistics constrain the product of electronic, vibrational, rotational and nuclear-spin exchange symmetries. In the ground electronic and vibrational states of these homonuclear molecules, the first two factors are symmetric. Exchanging the nuclei reverses the internuclear direction, so a rotational state has parity $(-1{)}^J$.

For ${}^1{\mathrm{H}}_2$, the protons are spin-one-half fermions and the total nuclear-exchange wavefunction must be antisymmetric. The antisymmetric spin singlet pairs with even $J$ (para-hydrogen, one spin state); the symmetric spin triplet pairs with odd $J$ (ortho-hydrogen, three spin states). For ${}^2{\mathrm{H}}_2$, spin-one deuterons are bosons, so the total must be symmetric. Nuclear spins $I=0,2$ supply six symmetric states paired with even $J$ (ortho-deuterium); $I=1$ supplies three antisymmetric states paired with odd $J$ (para-deuterium). The high-temperature ortho/para statistical ratios are $3:1$ for hydrogen and $2:1$ for deuterium, while low-temperature equilibrium populations also depend on rotational energies and whether spin conversion can occur.