Ro-vibronic species, overall allowed species, and nuclear spin statistical weights for symmetric top molecules belonging to the D n d and D n h ( n ≤6) point-groups

Nuclear spin statistical weight factors have been calculated for the rotational energy levels of symmetric top molecules belonging to the Dnd (n=2,3,4,5,6) and Dnh (n=3,4,5,6) point-groups. The ro-vibronic species for the levels have been calculated on the basis of Hougen’s classification scheme, using the point-group of the molecule. The Longuet–Higgins permutation–inversion-group was used to determine the overall allowed species permitted by the Pauli exclusion principle and the ramifications of inversion symmetry. The ro-vibronic species and the statistical weights are given in a set of easy to use tables. They are presented explicitly for all possible vibrational states of a totally symmetric electronic state. The weights are given in terms of the coefficients mα which occur in the expression ΓS=ΣmαΓαG for the reduction of the nuclear spin representation ΓS with respect to the irreducible representations ΓαG of the point-group G. General formulas for the characters of the nuclear spin representation are derived. These permit the numerical evaluation of the coefficients mα for a specific molecule by means of the standard group theoretical reduction formula. The results are general and are applicable to any molecule belonging to one of the symmetric top point-groups Dnd and Dnh with n?6, having an arbitrary number of atoms of the same or different kind, and for arbitrary values of the nuclear spins. The tables for the ro-vibronic species and for the statistical weights are arranged in such a manner that account can be taken of the splittings of level degeneracies due to rotation–vibration interaction. Extensions of the tables to include cases of nontotally symmetric electronic states are described. A few examples are worked out for molecules of increasing complexity and belonging to different point-groups to illustrate the ease of using the tables.