Symmetry considerations for internal rotation in ethylene-like molecules

The theory of the symmetry properties of the rotational and torsional levels is developed for molecules consisting of two identical XY₂ groups connected by a symmetrical linear chain of atoms, with an arbitrary barrier to internal rotation. The levels are classified according to the representations of the double group $\text{G}{}_{16}{}^{\mathrm{(2)}}$ of the Longuet-Higgins permutation-inversion group, and selection rules for electric dipole transitions are derived. It is shown that the symmetries of the normal coordinates of ethylene-like molecules can be different from those given recently by Papou?ek, Sarka, ?pirko and Jordanov. The results are applicable to the vibrational spectra of molecules like B₂F₄ and B₂Cl₄, which have low barriers to internal rotation in their ground states, and to the vibrational and rotational structure of electronic transitions of C₂H₄, where the combining electronic states may have very different torsional potential functions.