Species of the D2h Point Group

The D2h point group has the symmetry elements /, (Ti, 0-2, 0-3, Cf, and three C2 s. However, only three of these are considered necessary, since the others can all be obtained by the performance of two of the three necessary symmetry operations in succession. The number of necessary symmetry elements can be used to calculate the number of species a D2h group will have. If we consider that for the three necessary elements the motion of the atoms can either be symmetric ( + ) or antisymmetric ( —), then there are only eight ways groups of three -h or — signs can be arranged. These are 

There are therefore only eight species of vibration for the Z 2/, group  

The molecule C2H4 belongs to the Z 2/, group, and has twelve (3N — 6) fundamentals, distributed as follows three (Raman active), one (inactive), two big (Raman active), one b (infrared active), one 62 (Raman active), two b2 (infrared active), and two (infrared active). In order to be brief, we shall not discuss each vibrational species in detail. However by analyzing the character table for the 21, group given in the Appendix, the reader can readily ascertain the species which are symmetric or antisymmetric with respect to each symmetry element. 

The entries for species e and f in character tables will not be + 1 or — 1 as it was for a and b species. The symmetry of degenerate vibrations is discussed in terms of the symmetry each component of the degeneracy has with respect to the symmetry elements. The entries can be generally explained as follows First, for doubly degenerate vibrations, if for the symmetry element one of the components of the vibration is symmetric (+1) and the second antisymmetric (—1), the entry is 0 if both components are symmetric, the entry is -h2 and if both are antisymmetric, the entry is —2. 

If the vibration is triply degenerate the entries can be -h3 if all three components are symmetric, — 3 if all are antisymmetric, -h 1 if 

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