STATISTICAL WEIGHTS FOR THE ROTATIONAL LEVELS OF RIGID SYMMETRIC TOP MOLECULES

(1). E. Br. Wilson, JR., J. Chem. Phys. 3, 276 (1935).Author Institution:Several reports in the recent literature on rotation-vibration spectra rely on the use of the nuclear spin statistical weights of the rotational levels of polyatomic molecules. In some cases newly cited weights are found to be in error, or available weights which are correct for one molecule of a given point group are incorrectly assumed to be valid for different molecule of the same point group or, for want of available weights, reasonable assumptions about them were made in other cases for the analysis of partially resolved rotation-vibration bands. In this work we provide simple working formulae and tables which give these statistical weights for molecules which are characterized solely be their point group symmetry and for arbitrary values of the nuclear spins, augmented by clearly laid out procedure which allow a simple numerical evaluation of the general formulae for the case of specific molecule. We have used the method of Wilson (1) to derive a set of general tables and associated formulae for the spin characters for the nuclear spin statistical weights of the fundamental rotation-vibrational energy levels of rigid symmetric top molecules which are in a totally symmetric electronic state and which belong to the point groups $D_{nh} (n= 3,4,5,6,)$ and $D_{nd} (n= 2,3,4,5,6,)$. The weights are expressed in terms of the coefficients $m_{\alpha}$ of the reduced spin representation $\Gamma_{s} = \Sigma m_{\alpha} \Gamma_{\alpha}$ where the $\Gamma_{\alpha}$ are the irreducible representations of the rotational subgroup of the appropriate full molecular point group. Formulae are given for the spin characters, $X_{s}$, for arbitrary values of the nuclear spins. These permit a simple arithmetic evaluation of the coefficients $m_{\alpha}$. For each point group we also give a simple rule which permits, for a given molecule, the determination of the allowed species consistent with the pauli principle. The method of derivation of the tables and formulae will be demonstrated by means of the $D_{3h}$ point group, and the use of the results will be illustrated by means of worked out examples for several $D_{3h}$ molecules of increasing complexity