Group theoretical treatment of the planar internal rotation problem in (HF)2

The HF dimer is believed to exhibit an internal rotation tunneling process between two planar but nonlinear equilibrium configurations, during which tunneling the roles of the hydrogen-bonded and the free hydrogen atom are interchanged. This process can be represented schematically with labeled atoms as H_lF_aH₂F_b ? F_aH_lF_bH₂, and gives rise to a permutation-inversion group G₄ containing four operations. In the present work the vibration-rotation-tunneling problem in (HF)₂ is treated group theoretically in three ways: (i) by allowing tunneling only through a trans planar C_2h intermediate, (ii) by allowing tunneling only through a cis planar C_2v intermediate, and (iii) by considering the trans and cis tunneling processes both to occur, though not necessarily with the same probability. The molecular symmetry groups used for these treatments are (i) the point group C_2h, (ii) the point group C_2v, and (iii) a double group, which might be thought of as $\text{G}{}_4{}^2\text{= C}{}_{\mathrm{2h}}{}^2\text{= C}{}_{\mathrm{2v}}{}^2$. Nonplanar tunneling paths are not considered, since the internal axis method (IAM) coordinate system used here cannot easily be adapted to nonplanar internal rotation motions in this molecule. Various-details of energy level diagrams, symmetry species for operators, selection rules for spectroscopic transitions, and statistical weights are presented for the (HF)₂ tunneling problem, as well as some speculation on the general question of when point groups, permutation-inversion groups, or double groups are preferable for treating large-amplitude vibrational motion problems.