Tables and Types Species of Vibrations

The character table of a point group can be derived in a rigorous manner using the concepts of matrix algebra and the geometric model of a molecule. Since this is not the purpose of this text, the following 

The number of species is not great and it will be of value to discuss the more important ones. In Table 4-II, the species found for each point group are listed together with the symmetry elements. Most of the character tables are presented in the Appendix, and it is from these character tables that the qualitative explanation presented below was obtained. 

The accepted notation for species of vibration employs lowercase symbols if the vibration is a fundamental and capital letters if it is an overtone or combination band. However, in general discussions of species, we shall use the lower-case symbols consistently. In Table 4-II the species are listed in this manner. 

For the point group only one species of vibration a for the normal vibrations is listed since no symmetry element exists for molecules except the identity I. 

Each of the groups C2, Q, and has one symmetry element C2 has one twofold axis, Q has one mirror plane, and C,- has an inversion center. For all three of these groups there are two species, one of which is symmetric with respect to the element of symmetry and one of which is antisymmetric. The symmetric species for C2, Q, and Ci are designated as a, a, and Ug, respectively, and the antisymmetric species as b, a , and a. The subscripts g and u will be found on all species symbols of groups with a center of symmetry element, the g always indicating symmetric and the u antisymmetric vibrations relative to the inversion center. 

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