LESS CONFUSION AND MORE INFORMATION IN NOTATION FOR SYMMETRY GROUPS OF MOLECULES WITH LAMs

Abstract

Many molecular symmetry (MS) groups for molecules with periodic large-amplitude motions are semidirect products $H \land F$ where the invariant subgroup $H$ is a pure permutation group and $F$ is a point group.\footnote{C.M. Woodman, Mol. Phys. (1970) 19, 753-780} Group $H$ is often a direct product of cyclic permutation groups. The point group $F$ is formed by the equivalent rotations and, for starred PI operators, the $E^*$ operator. In a modified notation for such MS groups based on the semidirect product decomposition,\footnote{P. Groner, J. Mol. Spectrosc. (2018) 343, 34-43} such as $G_8 =[4]C_{s}$ or $G_{36} = [33]C_{2v}$, the numbers within square brackets refer to the cyclic operators in $H$ and the point group symbol refers to $F$. This has many advantages over just $G_{8}$ or $G_{36}$ because (1) it contains the number of periodic internal motions and their periodicities; (2) the symmetry of rotational, vibrational and transition dipole operators are the same as those of the point group; (3) this makes it easy to correlate the symmetry labels to those of possible equilibrium structures and vibronic levels; (4) and to derive qualitative splitting patterns. Each of these advantages is illustrated with examples. The notation can be extended to molecules with coaxial internal rotors on a linear frame. For such molecules with inequivalent rotors, two different valid notations are possible. If the rotors are symmetrically equivalent, the semidirect product corresponds to the extended PI (or EMS) group.