The symmetry groups of linear molecules

The ideas of Hougen and Longuet-Higgins concerning molecular symmetry have been extended to linear molecules by introducing "extended permutation-inversion groups". Using such groups it is possible to show that the rovibronic states (rotational levels) of a linear molecule can be classified according to the symmetry species of the point group ( D_{∞ h} or C_{∞ v}). For a molecule with D_{∞ h} symmetry the rotational levels can be labeled Σ_g⁺, Σ_u⁺, Σ_g^-, or Σ_u^-, which is equivalent to the usual notation of + s, + a, - a, or - s respectively. For a molecule with C_{∞ v} symmetry the rotational levels can be labeled Σ ⁺ or Σ ^-, which is equivalent to the usual notation of + or -, respectively. Although no new symmetry predictions are obtained this does remove an obscurity concerning the symmetry classification of the levels of a linear molecule. Further-more, there is an interesting group-theoretical relationship between the extended permutation-inversion group of a linear molecule (which is isomorphic with D_{∞ h} or C_{∞ v}) and the corresponding permutation-inversion group (which is isomorphic with C_{2 v} or C_s).