Computer-generated character tables and nuclear spin statistical weights: application to benzene dimer and methane dimer

In the assignment of high-resolution spectra of van der Waals molecular clusters, it is of great utility to know the character table of the molecular symmetry group and the nuclear spin weights of the various rovibronic symmetry species [1]. While their calculation is not difficult, for large groups it is lengthy [2] and prone to errors due to the sheer quantity of variables involved. For the benzene dimer, both the character table [3] and the nuclear spin weights [4] have been published with errors (though the latter had previously been published correctly [5]); for methane dimer, the nuclear spin weights have been published with an error [2]. We would like to bring to the attention of the spectroscopy community a free software package for group theory named GAP [6], which greatly facilitates these calculations. As an example of its usage, we present calculations for the permutation–inversion (PI) groups of benzene dimer, which is the direct product of a permutation (P) group and the inversion group fE;E g, and of methane dimer, for which a planar structure is not accessible and thus, E is not a feasible operation. For both dimers, rigid monomer units are assumed. Calculations for benzene and methane trimers, and water hexamer, can be done in this way within a few seconds on a personal computer. In what follows, we consider only the hydrogen atoms, since the C isotope has nuclear spin zero. We label the hydrogen atoms on the two benzene monomers 1. . .6 and 7. . .12, in a circular way. The group of feasible permutations is specified by its generators, for example C6, C0 2, and s (monomer exchange), by entering at the GAP prompt: